RD Sharma solutions for Class 12 Maths chapter 5 - Algebra of Matrices [Latest edition]

Chapters Chapter 5: Algebra of Matrices

Exercise 5.1Exercise 5.2Exercise 5.3Exercise 5.4Exercise 5.5Exercise 5.6Exercise 5.7
Exercise 5.1 [Pages 5 - 8]

RD Sharma solutions for Class 12 Maths Chapter 5 Algebra of MatricesExercise 5.1 [Pages 5 - 8]

Exercise 5.1 | Q 1 | Page 6

If a matrix has 8 elements, what are the possible orders it can have? What if it has 5 elements?

Exercise 5.1 | Q 2 | Page 6

If A = [aij] =[[2,3,-5],[1,4,9],[0,7,-2]]and B = [bij] [[2,-1],[-3,4],[1,2]]

then find (i) a22 + b21 (ii) a11 b11 + a22 b22

Exercise 5.1 | Q 3 | Page 6

Let A be a matrix of order 3 × 4. If R1 denotes the first row of A and C2 denotes its second column, then determine the orders of matrices R1 and C2

Exercise 5.1 | Q 4.1 | Page 7

Construct a 2 × 3 matrix whose elements aij are given by :

(i) aij = j

Exercise 5.1 | Q 4.2 | Page 7

Construct a 2 × 3 matrix whose elements aij are given by :

(ii) aij = 2i − j

Exercise 5.1 | Q 4.3 | Page 7

Construct a 2 × 3 matrix whose elements aij are given by :

(iii) aij = i + j

Exercise 5.1 | Q 4.4 | Page 7

Construct a 2 × 3 matrix whose elements aij are given by :

(iv) aij =(i+j)^2/2

Exercise 5.1 | Q 5.1 | Page 7

Construct a 2 × 2  matrix whose elements a_(ij)

are given by: (i+j)^2/2

Exercise 5.1 | Q 5.2 | Page 7

Construct a 2 × 2 matrix whose elements aij are given by:

aij=(i-j)^2/2

Exercise 5.1 | Q 5.3 | Page 7

Construct a 2 × 2 matrix whose elements aij are given by:

a_(ij)=(i-2_j)^2/2

Exercise 5.1 | Q 5.4 | Page 7

Construct a 2 × 2 matrix whose elements aij are given by:

a_(ij)= (2i +j)^2/2

Exercise 5.1 | Q 5.5 | Page 7

Construct a 2 × 2 matrix whose elements aij are given by:

a_(ij)=|2_i - 3_i|/2

Exercise 5.1 | Q 5.6 | Page 7

Construct a 2 × 2 matrix whose elements aij are given by:

a_(ij)=|-3i +j|/2

Exercise 5.1 | Q 5.7 | Page 7

Construct a 2 × 2 matrix whose elements aij are given by:

a_(ij)=e^(2ix) sin (xj)

Exercise 5.1 | Q 6.1 | Page 7

Construct a 3 × 4 matrix A = [aij] whose elements aij are given by:

aij i + j

Exercise 5.1 | Q 6.2 | Page 7

Construct a 3 × 4 matrix A = [aij] whose elements aij are given by:

aij = i −

Exercise 5.1 | Q 6.3 | Page 7

Construct a 3 × 4 matrix A = [aij] whose elements aij are given by:

aij = 2i

Exercise 5.1 | Q 6.4 | Page 7

Construct a 3 × 4 matrix A = [aij] whose elements aij are given by:

aij = j

Exercise 5.1 | Q 6.5 | Page 7

Construct a 3 × 4 matrix A = [aij] whose elements aij are given by:

a_(ij)=1/2= -3i + j

Exercise 5.1 | Q 7.1 | Page 7

Construct a 4 × 3 matrix whose elements are

a_(ij)=2_i+ i/j

Exercise 5.1 | Q 7.2 | Page 7

Construct a 4 × 3 matrix whose elements are

a_(ij)= (i-j)/(i+j )

Exercise 5.1 | Q 7.3 | Page 7

Construct a 4 × 3 matrix whose elements are

aij =

Exercise 5.1 | Q 8 | Page 5

find x, y , a and b if  $\begin{bmatrix}3x + 4y & 2 & x - 2y \\ a + b & 2a - b & - 1\end{bmatrix} = \begin{bmatrix}2 & 2 & 4 \\ 5 & - 5 & - 1\end{bmatrix}$

Exercise 5.1 | Q 9 | Page 7

Find xya and b if

[[2x-3y,a-b,3],[1,x+4y,3a+4b]]=[[1,-2,3],[1,6,29]]

Exercise 5.1 | Q 10 | Page 7

Find the values of abc and d from the following equations:[[2a + b,a-2b],[5c-d,4c + 3d ]]= [[4,- 3],[11,24]]

Exercise 5.1 | Q 11 | Page 7

Find xy and z so that A = B, whereA= [[x-2,3,2x],[18z,y+2,6x]],b=[[y,z,6],[6y,x,2y]]

Exercise 5.1 | Q 12 | Page 7

If [[x,3x- y],[2x+z,3y -w ]]=[[3,2],[4,7]] find x,y,z,w

Exercise 5.1 | Q 13 | Page 7

If [[x-y,z],[2x-y,w]]=[[-1,4],[0,5]]Find X,Y,Z,W.

Exercise 5.1 | Q 14 | Page 7

If [[x + 3 , z + 4 ,     2y-7 ],[4x + 6,a-1,0 ],[b-3,3b,z + 2c ]]= [[0,6,3y-2],[2x,-3,2c-2],[2b + 4,-21,0]]Obtain the values of abcxy and z.

Exercise 5.1 | Q 15 | Page 8

If [[2x +1    5x],[0     y^2 +1]]= [[x+3   10],[0      26 ]], find the value of (x + y).

Exercise 5.1 | Q 16 | Page 8

If [[xy          4],[z+6     x+y ]]=[[8     w],[0     6]], then find the values of X,Y,Z and W .

Exercise 5.1 | Q 17.1 | Page 8

Given an example of

a row matrix which is also a column matrix,

Exercise 5.1 | Q 17.2 | Page 8

Given an example of

a diagonal matrix which is not scalar,

Exercise 5.1 | Q 17.3 | Page 8

Given an example of

a triangular matrix

Exercise 5.1 | Q 18 | Page 8

The sales figure of two car dealers during January 2013 showed that dealer A sold 5 deluxe, 3 premium and 4 standard cars, while dealer B sold 7 deluxe, 2 premium and 3 standard cars. Total sales over the 2 month period of January-February revealed that dealer A sold 8 deluxe 7 premium and 6 standard cars. In the same 2 month period, dealer B sold 10 deluxe, 5 premium and 7 standard cars. Write 2 × 3 matrices summarizing sales data for January and 2-month period for each dealer.

Exercise 5.1 | Q 19 | Page 8

For what values of x and y are the following matrices equal?

A=[[2x+1   2y],[0              y^2 - 5y]]B=[[x + 3      y^2 +2],[0        -6]]

Exercise 5.1 | Q 20 | Page 8

Find the values of x and y if

[[X + 10,Y^2 + 2Y],[0, -4]]=[[3x +4,3],[0,y^2-5y]]

Exercise 5.1 | Q 21 | Page 8

For what values of a and b if A = B, where

A = [[a + 4        3b],[8        -6]]   B = [[2a +2              b^2+2],[8                    b^2  - 5b]]

Disclaimer: There is a misprint in the question, b2 − 5should be written instead of b2 − 56.

Exercise 5.2 [Pages 18 - 19]

RD Sharma solutions for Class 12 Maths Chapter 5 Algebra of MatricesExercise 5.2 [Pages 18 - 19]

Exercise 5.2 | Q 1.1 | Page 18

Compute the following sums:

[[3   -2],[1           4]]+ [[-2         4 ],[1           3]]

Exercise 5.2 | Q 1.2 | Page 18

Compute the following sums:

[[2    1   3],[0   3   5],[-1   2   5]]+ [[1 -2     3],[2            6        1],[0   -3       1]]

Exercise 5.2 | Q 2.1 | Page 18

Let A = [[2,4],[3,2]], B=[[1,3],[-2,5]]and c =[[-2,5],[3,4]].Find each of the following: 2A − 3B

Exercise 5.2 | Q 2.2 | Page 18

Let A = [[2,4],[3,2]], B=[[1,3],[-2,5]]and c =[[-2,5],[3,4]].Find each of the following:  B − 4C

Exercise 5.2 | Q 2.3 | Page 18

Let A = [[2,4],[3,2]], B=[[1,3],[-2,5]]and c =[[-2,5],[3,4]].Find each of the following: 3A − C

Exercise 5.2 | Q 2.4 | Page 18

Let A = [[2,4],[3,2]], B=[[1,3],[-2,5]]and c =[[-2,5],[3,4]].Find each of the following: 3A − 2B + 3C

Exercise 5.2 | Q 3.1 | Page 18

If A =[[2,3],[5,7]],B = [[-1,0 ,2],[3,4,1]],C= [[-1,2,3],[2,1,0]]find : A + B and B + C

Exercise 5.2 | Q 3.2 | Page 18

If A =[[2   3],[5   7]],B = [[-1   0   2],[3    4      1]],C= [[-1    2   3],[2    1     0]]find

2B + 3A and 3C − 4B

Exercise 5.2 | Q 4 | Page 18

Let A = [[-1    0    2],[3     1      4]]B=[[0      -2     5],[1      -3     1]]and C = [[1     -5       2],[6     0    -4 ]]Compute2A2-3B +4C :

Exercise 5.2 | Q 5.1 | Page 18

If A = diag (2 − 59), B = diag (11 − 4) and C = diag (−6 3 4), find: A − 2B

Exercise 5.2 | Q 5.2 | Page 18

If A = diag (2 − 59), B = diag (11 − 4) and C = diag (−6 3 4), find

B + C − 2A

Exercise 5.2 | Q 5.3 | Page 18

If A = diag (2 − 59), B = diag (11 − 4) and C = diag (−6 3 4), find

2A + 3B − 5C

Exercise 5.2 | Q 6 | Page 18

Given the matrices

A=[[2,1,1],[3,-1,0],[0,2,4]] , B=[[9,7,-1],[3,5,4],[2,1,6]]  and  C=[[2,-4,3],[1,-1,0],[9,4,5]]

Verify that (A + B) + C = A + (B + C).

Exercise 5.2 | Q 7 | Page 18

Find matrices X and Y, if X + Y =[[5     2],[0       9]]

and X − Y =  [[3       6],[0   -1]]

Exercise 5.2 | Q 8 | Page 18

Find X if Y =[[3       2],[1      4]]and 2X + Y =[[1       0],[-3        2]]

Exercise 5.2 | Q 9 | Page 18

Find matrices X and Y, if 2X − Y = [[6       -6           0],[-4            2           1]]and X + 2Y =[[3              2                     5],[-2         1    -7 ]]

Exercise 5.2 | Q 10 | Page 18

X − Y =[[1      1       1],[1        1          0],[1         0          0]] and X + Y = [[3        5         1],[-1       1           1],[11       8           0]]find X and Y.

Exercise 5.2 | Q 11 | Page 18

Find matrix A, if  [[1         2      -1],[0         4       9]]

+ A = [[9        -1           4],[-2        1            3]]

Exercise 5.2 | Q 12 | Page 18

If A =[[9     1],[7      8]],B=[[1      5],[7      12]]find matrix C such that 5A + 3B + 2C is a null matrix.

Exercise 5.2 | Q 13 | Page 18

If A = [[2      -2],[4             2],[-5          1]],B=[[8             0],[4      -2],[3          6]]

, find matrix X such that 2A + 3X = 5B.

Exercise 5.2 | Q 14 | Page 18

If A = [[1    -3         2],[2        0               2]]and B = [[2          -1           -1],[1           0             -1]] find the matrix C such that A + B + C is

, find the matrix C such that A + B + C is zero matrix.

Exercise 5.2 | Q 15.1 | Page 18

Find xy satisfying the matrix equations

[[X-Y               2            -2],[4                        x                6]]+[[3        -2                2],[1         0            -1]]=[[                6                       0                             0],[         5                       2x+y                5]]

Exercise 5.2 | Q 15.2 | Page 18

Find xy satisfying the matrix equations

[x     y + 2    z-3 ] +  [  y       4          5]=[4        9        12]

Exercise 5.2 | Q 15.3 | Page 18

Find xy satisfying the matrix equations

x[,]+y[,]+[[-8],[-11]]=0

Exercise 5.2 | Q 16 | Page 19

If 2 [[3    4],[5     x]]+[[1   y],[0    1]]=[[7        0],[10      5]] find x and y.

Exercise 5.2 | Q 17 | Page 19

Find the value of λ, a non-zero scalar, if λ

Exercise 5.2 | Q 18.1 | Page 19

Find a matrix X such that 2A + B + X = O, where

A= [[-1      2],[3        4]],B= [[3       -2],[1          5]]

Exercise 5.2 | Q 18.2 | Page 19

Find a matrix X such that 2A + B + X = O, where

If A = [[8            0],[4    -2],[3         6]] and B = [[2       -2],[4           2],[-5          1]]

, then find the matrix X of order 3 × 2 such that 2A + 3X = 5B.

Exercise 5.2 | Q 19.1 | Page 19

Find xyz and t, if

3[[x     y],[z      t]]=[[x        6],[-1          2t]]+[[4             x+y],[z+t         3]]

Exercise 5.2 | Q 19.2 | Page 19

Find xyz and t, if

2[[x         5],[z         t]]+[[x           6],[-1          2t]]=[[7            14],[15        14]]

Exercise 5.2 | Q 20 | Page 19

If X and Y are 2 × 2 matrices, then solve the following matrix equations for X and Y.

2X + 3Y = [[2,3],[4,0]], 3X+2Y = [[-2,2],[1,-5]]

Exercise 5.2 | Q 21 | Page 19

In a certain city there are 30 colleges. Each college has 15 peons, 6 clerks, 1 typist and 1 section officer. Express the given information as a column matrix. Using scalar multiplication, find the total number of posts of each kind in all the colleges.

Exercise 5.2 | Q 22 | Page 19

The monthly incomes of Aryan and Babban are in the ratio 3 : 4 and their monthly expenditures are in the ratio 5 : 7. If each saves Rs 15,000 per month, find their monthly incomes using matrix method. This problem reflects which value?

Exercise 5.3 [Pages 41 - 48]

RD Sharma solutions for Class 12 Maths Chapter 5 Algebra of MatricesExercise 5.3 [Pages 41 - 48]

Exercise 5.3 | Q 1.1 | Page 41

Compute the indicated products:

[[a    b],[-b      a]][[a     -b],[b         a]]

Exercise 5.3 | Q 1.2 | Page 41

Compute the indicated products:

[[1     -2],[2     3]][[1         2        3],[-3    2      -1]]

Exercise 5.3 | Q 1.3 | Page 41

Compute the indicated products:

[[2      3      4],[3      4        5],[4        5         6]][[1       -3        5],[0          2         4],[3           0            5]]

Exercise 5.3 | Q 2.1 | Page 41

Show that AB ≠ BA in each of the following cases:

A= [[5    -1],[6        7]]And B =[[2       1],[3         4]]

Exercise 5.3 | Q 2.2 | Page 41

Show that AB ≠ BA in each of the following cases

A=[[-1          1           0],[0          -1           1],[2                  3                4]]  and  =B [[1          2            3], [0          1           0],[1        1          0]]

Exercise 5.3 | Q 2.3 | Page 41

Show that AB ≠ BA in each of the following cases:

A=[[1       3         0],[1        1          0],[4         1         0]]And    B=[[0      1          0],[1        0        0],[0           5          1]]

Exercise 5.3 | Q 3.1 | Page 41

Compute the products AB and BA whichever exists in each of the following cases:

A= [[1      -2],[2              3]] and  B=[[1       2        3],[2         3         1]]

Exercise 5.3 | Q 3.2 | Page 41

Compute the products AB and BA whichever exists in each of the following cases:

A=[[3     2],[-1     0],[-1      1]] and B= [[4         5        6],[0           1             2]]

Exercise 5.3 | Q 3.3 | Page 41

Compute the products AB and BA whichever exists in each of the following cases:

A = [1 −1 2 3] and B=[,,,]

Exercise 5.3 | Q 3.4 | Page 41

Compute the products AB and BA whichever exists in each of the following cases:

[ab][[c],[d]]+ [a, b, c, d] [[a],[b],[c],[d]]

Exercise 5.3 | Q 4.1 | Page 41

Show that AB ≠ BA in each of the following cases:

A = [[1,3,-1],[2,-1,-1],[3,0,-1]] And B= [[-2,3,-1],[-1,2,-1],[-6,9,-4]]

Exercise 5.3 | Q 5.1 | Page 41

Evaluate the following:

([[1              3],[-1    -4]]+[[3        -2],[-1         1]])[[1         3           5],[2            4               6]]

Exercise 5.3 | Q 5.2 | Page 41

Evaluate the following:

[[],[1  2  3],[]] [[1     0      2],[2       0         1],[0          1       2]] [,,]

Exercise 5.3 | Q 5.3 | Page 41

Evaluate the following:

[[1     -1],[0            2],[2           3]]  ([[1     0        2],[2        0        1]]-[[0             1                 2],[1           0                    2]])

Exercise 5.3 | Q 6 | Page 41

If A = [[1     0],[0        1]],B[[1            0],[0       -1]]

and C= [[0      1],[1       0]]

, then show that A2 = B2 = C2 = I2.

Exercise 5.3 | Q 7 | Page 42

If A = [[2       -1],[3             2]]  and B = [[0         4],[-1          7]]find 3A2 − 2B + I

Exercise 5.3 | Q 8 | Page 42

If A =  [[4       2],[-1        1]]

, prove that (A − 2I) (A − 3I) = O

Exercise 5.3 | Q 9 | Page 42

If A =  [[1    1],[0    1]]  show that A2 = [[1       2],[0          1]] and A3 = [[1        3],[0       1]]

Exercise 5.3 | Q 10 | Page 42

If A = [[ab,b^2],[-a^2,-ab]] , show that A2 = O

Exercise 5.3 | Q 11 | Page 42

If A = [[ cos 2θ     sin 2θ],[ -sin 2θ    cos 2θ]], find A2.

Exercise 5.3 | Q 12 | Page 42

If A =

$\begin{bmatrix}2 & - 3 & - 5 \\ - 1 & 4 & 5 \\ 1 & - 3 & - 4\end{bmatrix}$and B =

$\begin{bmatrix}- 1 & 3 & 5 \\ 1 & - 3 & - 5 \\ - 1 & 3 & 5\end{bmatrix}$ , show that AB = BA = O3×3.

Exercise 5.3 | Q 13 | Page 42

If A = [[0,c,-b],[-c,0,a],[b,-a,0]]and B =[[a^2 ,ab,ac],[ab,b^2,bc],[ac,bc,c^2]], show that AB = BA = O3×3.

Exercise 5.3 | Q 14 | Page 42

If A =[[2     -3          -5],[-1             4           5],[1           -3       -4]] and B =[[2         -2            -4],[-1               3                  4],[1            2           -3]]

, show that AB = A and BA = B.

Exercise 5.3 | Q 15 | Page 42

Let A =[[-1            1               -1],[3         -3           3],[5           5             5]]and B =[[0                4                  3],[1              -3              -3],[-1               4                 4]]

, compute A2 − B2.

Exercise 5.3 | Q 16.1 | Page 42

For the following matrices verify the associativity of matrix multiplication i.e. (AB) C = A(BC):

A =-[[1             2         0],[-1        0           1]],B=[[1       0],[-1        2],[0        3]] and C= [,[-1]]

Exercise 5.3 | Q 16.2 | Page 42

For the following matrices verify the associativity of matrix multiplication i.e. (ABC = A(BC):

A=[[4       2        3],[1       1          2],[3         0          1]]=B=[[1        -1          1],[0         1            2],[2           -1          1]] and  C= [[1       2       -1],[3       0         1],[0         0         1]]

Exercise 5.3 | Q 17.1 | Page 42

For the following matrices verify the distributivity of matrix multiplication over matrix addition i.e. A (B + C) = AB + AC:

A = [[1     -1],[0          2]] B=   [[-1       0],[2        1]]and C= [[0       1],[1     -1]]

Exercise 5.3 | Q 17.2 | Page 42

For the following matrices verify the distributivity of matrix multiplication over matrix addition i.e. A (B + C) = AB + AC:

A=[[2    -1],[1        1],[-1         2]] B=[[0     1],[1      1]] C=[[1      -1],[0                1]]

Exercise 5.3 | Q 18 | Page 42

If A= [[1        0           -2],[3        -1           0],[-2              1               1]] B=,[[0         5           -4],[-2          1             3],[-1          0              2]] and  C=[[1               5              2],[-1           1              0],[0          -1             1]] verify that A (B − C) = AB − AC.

Exercise 5.3 | Q 19 | Page 43

Compute the elements a43 and a22 of the matrix:A=[[0     1        0],[2      0        2],[0       3        2],[4        0       4]] [[2       -1],[-3           2],[4              3]]  [[0            1           -1                    2                     -2],[3       -3             4          -4                  0]]

Exercise 5.3 | Q 20 | Page 43

$A = \begin{bmatrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ p & q & r\end{bmatrix}$ ,and I is the identity matrix of order 3, show that A3 = pI + qA +rA2.
Exercise 5.3 | Q 21 | Page 43

If w is a complex cube root of unity, show that

([[1         w          w^2],[w            w^2             1],[w^2           1             w]]+[[w          w^2          1],[w^2             1               w],[w            w^2              1]])[,[w],[w^2]]=[,,]

Exercise 5.3 | Q 22 | Page 43

$A = \begin{bmatrix}2 & - 3 & - 5 \\ - 1 & 4 & 5 \\ 1 & - 3 & - 4\end{bmatrix}$   , Show that A2 = A.

Exercise 5.3 | Q 23 | Page 43

If  $A = \begin{bmatrix}4 & - 1 & - 4 \\ 3 & 0 & - 4 \\ 3 & - 1 & - 3\end{bmatrix}$     ,  Show that A2 = I3.

Exercise 5.3 | Q 24.1 | Page 43

If [1 1 x] [[1         0            2],[0           2         1],[2            1           0]] [,,] = 0, find x.

Exercise 5.3 | Q 24.2 | Page 43

If [[2     3],[5      7]] [[1      -3],[-2       4]]-[[-4      6],[-9        x]] find x.

Exercise 5.3 | Q 25 | Page 42

If [x 4 1] [[2       1          2],[1         0          2],[0       2 -4]]  [[x],,[-1]] = 0, find x.

Exercise 5.3 | Q 26 | Page 43

If [1 −1 x] [[0       1           -1],[2           1             3],[1          1             1]]   [,,]== 0, find x.

Exercise 5.3 | Q 27 | Page 43

$A = \begin{bmatrix}3 & - 2 \\ 4 & - 2\end{bmatrix} and \text{ I }= \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$,  then prove that A2 − A + 2I = O.

Exercise 5.3 | Q 28 | Page 43

$A = \begin{bmatrix}3 & 1 \\ - 1 & 2\end{bmatrix} and \text{ I} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$

Exercise 5.3 | Q 29 | Page 43

If

Exercise 5.3 | Q 30 | Page 43

$A = \begin{bmatrix}2 & 3 \\ - 1 & 0\end{bmatrix}$,show that A2 − 2A + 3I2 = O

Exercise 5.3 | Q 31 | Page 43

Show that the matrix  $A = \begin{bmatrix}2 & 3 \\ 1 & 2\end{bmatrix}$satisfies the equation A3 − 4A2 + A = O

Exercise 5.3 | Q 32 | Page 43

Show that the matrix $A = \begin{bmatrix}5 & 3 \\ 12 & 7\end{bmatrix}$  is  root of the equation A2 − 12A − I = O

Exercise 5.3 | Q 33 | Page 43

If $A = \begin{bmatrix}3 & - 5 \\ - 4 & 2\end{bmatrix}$ , find A2 − 5A − 14I.

Exercise 5.3 | Q 34 | Page 44

$A = \begin{bmatrix}3 & 1 \\ - 1 & 2\end{bmatrix}$show that A2 − 5A + 7I = O use this to find A4.

Exercise 5.3 | Q 35 | Page 44

If A=, find k such that A2 = kA − 2I2

Exercise 5.3 | Q 36 | Page 44

If

Exercise 5.3 | Q 37 | Page 44

If$A = \begin{bmatrix}1 & 2 \\ 2 & 1\end{bmatrix}$ f (x) = x2 − 2x − 3, show that f (A) = 0

Exercise 5.3 | Q 38 | Page 44

If A=then find λ, μ so that A2 = λA + μI

Exercise 5.3 | Q 39 | Page 44

Find the value of x for which the matrix product[[2       0           7],[0          1            0],[1       -2       1]] [[-x         14x          7x],[0         1            0],[x           -4x             -2x]]equal an identity matrix.

Exercise 5.3 | Q 40.1 | Page 44

Solve the matrix equations:

[x1][[1,0],[-2,-3]][[x],]=0

Exercise 5.3 | Q 40.2 | Page 44

Solve the matrix equations:

[1  2   1] [[1,2,0],[2,0,1],[1,0 ,2]][,,[x]]=0

Exercise 5.3 | Q 40.3 | Page 44

Solve the matrix equations:

[[],[x-5-1],[]][[1,0,2],[0,2,1],[2,0,3]] [[x],,]=0

Exercise 5.3 | Q 40.4 | Page 44

Solve the matrix equations:

[2x 3] [[1       2],[-3      0]] , [[x],]=0

Exercise 5.3 | Q 41 | Page 44

If A= [[1,2,0],[3,-4,5],[0,-1,3]] compute A2 − 4A + 3I3.

Exercise 5.3 | Q 42 | Page 44

If f (x) = x2 − 2x, find f (A), where A=

Exercise 5.3 | Q 43 | Page 44

If f (x) = x3 + 4x2 − x, find f (A), where$A = \begin{bmatrix}0 & 1 & 2 \\ 2 & - 3 & 0 \\ 1 & - 1 & 0\end{bmatrix}$

Exercise 5.3 | Q 44 | Page 44

If , then show that A is a root of the polynomial f (x) = x3 − 6x2 + 7x + 2.

Exercise 5.3 | Q 45 | Page 44

A=[[1,2,2],[2,1,2],[2,2,1]], then prove that A2 − 4A − 5I = 0

Exercise 5.3 | Q 46 | Page 44

A=[[3,2, 0],[1,4,0],[0,0,5]] show that A2 − 7A + 10I3 = 0

Exercise 5.3 | Q 47 | Page 44

Without using the concept of inverse of a matrix, find the matrix [[x       y],[z       u]] such that
[[5     -7],[-2         3]][[x        y],[z         u]]=[[-16       -6],[7                   2]]

Exercise 5.3 | Q 48.1 | Page 45

Find the matrix A such that [[1     1],[0       1]]A=[[3        3         5],[1       0          1]]

Exercise 5.3 | Q 48.2 | Page 45

Find the matrix A such that A=[[1,2,3],[4,5,6]]=  [[-7,-8,-9],[2,4,6]]

Exercise 5.3 | Q 48.3 | Page 45

Find the matrix A such that [,,]  A=[[-4,8,4],[-1,2,1],[-3,6,3]]

Exercise 5.3 | Q 48.4 | Page 45

Find the matrix A such that    [2  1  3 ] [[-1,0,-1],[-1,1,0],[0,1,1]] [,,[-1]]=A

Exercise 5.3 | Q 48.5 | Page 45

Find the matrix A such that [[2,-1],[1,0],[-3,-4]]A =[[-1,-8,-10],[1,-2,-5],[9,22,15]]

Exercise 5.3 | Q 48.6 | Page 45

Find the matrix A such that =[[1,2,3],[4,5,6]]=[[-7,-8,-9],[2,4,6],[11,10,9]]

Exercise 5.3 | Q 49 | Page 45

Find a 2 × 2 matrix A such that A=[[1,-2],[1,4]]=6l_2

Exercise 5.3 | Q 50 | Page 45

If A=[[0,0],[4,0]] find A^16

Exercise 5.3 | Q 51 | Page 45

If A=[[0,-x],[x,0]],[[0,1],[1,0]] and x^2=-1, then  show that (A+B)^2=A^2+B^2

Exercise 5.3 | Q 52 | Page 45

A=[[1,0,-3],[2,1,3],[0,1,1]]then verify that A2 + A = A(A + I), where I is the identity matrix.

Exercise 5.3 | Q 53 | Page 45

A=[[3,-5],[-4,2]] then find A2 − 5A − 14I. Hence, obtain A3

Exercise 5.3 | Q 54.1 | Page 45

If P(x)=[[cos x,sin x],[-sin x,cos x]], then show that P(x),P(y)=P(x+y)=P(y)P(x).

Exercise 5.3 | Q 54.2 | Page 45

If P=[[x,0,0],[0,y,0],[0,0,z]] and Q=[[a,0,0],[0,b,0],[0,0,c]] prove that PQ=[[xa,0,0],[0,yb,0],[0,0,zc]]=QP

Exercise 5.3 | Q 55 | Page 45

A=[[2,0,1],[2,1,3],[1,-1,0]] , find A2 − 5A + 4I and hence find a matrix X such that A2 − 5A + 4I + = 0.

Exercise 5.3 | Q 56 | Page 45

If A=[[1,1],[0,1]] , Prove that A=[[1,n],[0,1]] for all positive integers n.

Exercise 5.3 | Q 57 | Page 45

If$A = \begin{bmatrix}a & b \\ 0 & 1\end{bmatrix}$, prove that$A^n = \begin{bmatrix}a^n & b( a^n - 1)/a - 1 \\ 0 & 1\end{bmatrix}$ for every positive integer n .

Exercise 5.3 | Q 58 | Page 45

If A=[[cos θ, i sinθ],[i sinθ,cosθ]] then prove by principle of mathematical induction that A^n=[[cos  nθ,i sinθ],[i sin nθ,cos nθ]] for all n  ∈ N.

Exercise 5.3 | Q 59 | Page 46

$A = \begin{bmatrix}\cos \alpha + \sin \alpha & \sqrt{2}\sin \alpha \\ - \sqrt{2}\sin \alpha & \cos \alpha - \sin \alpha\end{bmatrix}$ ,prove that

$A^n = \begin{bmatrix}\text{cos n α} + \text{sin n α} & \sqrt{2}\text{sin n α} \\ - \sqrt{2}\text{sin n α} & \text{cos n α} - \text{sin n α} \end{bmatrix}$ for all n ∈ N.

Exercise 5.3 | Q 60 | Page 46

Let A= [[1,1,1],[0,1,1],[0,0,1]] Use the principle of mathematical introduction to show  that A^n [[1,n,n(n+1)//2],[0,1,1],[0,0,1]] for every position integer n.

Exercise 5.3 | Q 61 | Page 46

If BC are n rowed square matrices and if A = B + CBC = CBC2 = O, then show that for every n ∈ NAn+1 = Bn (B + (n + 1) C).

Exercise 5.3 | Q 62 | Page 46

If A = diag (abc), show that An = diag (anbncn) for all positive integer n.

Exercise 5.3 | Q 63 | Page 46

If A is a square matrix, using mathematical induction prove that (AT)n = (An)T for all n ∈ ℕ.

Exercise 5.3 | Q 64 | Page 46

A matrix X has a + b rows and a + 2 columns while the matrix Y has b + 1 rows and a + 3 columns. Both matrices XY and YX exist. Find a and b. Can you say XY and YX are of the same type? Are they equal.

Exercise 5.3 | Q 65.1 | Page 46

Give examples of matrices
A and B such that AB ≠ BA

Exercise 5.3 | Q 65.2 | Page 46

Give examples of matrices

A and B such that AB = O but A ≠ 0, B ≠ 0.

Exercise 5.3 | Q 65.3 | Page 46

Give examples of matrices

A and B such that AB = O but BA ≠ O.

Exercise 5.3 | Q 65.4 | Page 46

Give examples of matrices

AB and C such that AB = AC but B ≠ CA ≠ 0.

Exercise 5.3 | Q 66 | Page 46

Let A and B be square matrices of the same order. Does (A + B)2 = A2 + 2AB + B2 hold? If not, why?

Exercise 5.3 | Q 67.1 | Page 46

If A and B are square matrices of the same order, explain, why in general

(A + B)2 ≠ A2 + 2AB + B2

Exercise 5.3 | Q 67.2 | Page 46

If A and B are square matrices of the same order, explain, why in general

(− B)2 ≠ A2 − 2AB + B2

Exercise 5.3 | Q 67.3 | Page 46

If A and B are square matrices of the same order, explain, why in general

(A + B) (A − B) ≠ A2 − B2

Exercise 5.3 | Q 68 | Page 46

Let A and B be square matrices of the order 3 × 3. Is (AB)2 = A2 B2? Give reasons.

Exercise 5.3 | Q 69 | Page 46

If A and B are square matrices of the same order such that AB = BA, then show that (A + B)2 = A2 + 2AB + B2.

Exercise 5.3 | Q 70 | Page 46

Let A=[[1,1,1],[3,3,3]],B=[[3,1],[5,2],[-2,4]] and C=[[4,2],[-3,5],[5,0]]Verify that AB = AC though B ≠ CA ≠ O.

Exercise 5.3 | Q 71 | Page 46

Three shopkeepers AB and C go to a store to buy stationary. A purchases 12 dozen notebooks, 5 dozen pens and 6 dozen pencils. B purchases 10 dozen notebooks, 6 dozen pens and 7 dozen pencils. C purchases 11 dozen notebooks, 13 dozen pens and 8 dozen pencils. A notebook costs 40 paise, a pen costs Rs. 1.25 and a pencil costs 35 paise. Use matrix multiplication to calculate each individual's bill.

Exercise 5.3 | Q 72 | Page 46

The cooperative stores of a particular school has 10 dozen physics books, 8 dozen chemistry books and 5 dozen mathematics books. Their selling prices are Rs. 8.30, Rs. 3.45 and Rs. 4.50 each respectively. Find the total amount the store will receive from selling all the items.

Exercise 5.3 | Q 73 | Page 46

In a legislative assembly election, a political group hired a public relations firm to promote its candidates in three ways: telephone, house calls and letters. The cost per contact (in paise) is given matrix A as

Cost per contact

A=[,,] [["Teliphone"] ,["House call "],[" letter"]]

The number of contacts of each type made in two cities X and Y is given in matrix B as

Telephone   House call    Letter

B= [[    1000, 500,      5000],[3000,1000,     10000                ]]

Find the total amount spent by the group in the two cities X and Y.

Exercise 5.3 | Q 74.1 | Page 47

A trust fund has Rs 30000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide Rs 30000 among the two types of bonds. If the trust fund must obtain an annual total interest of
(i) Rs 1800

Exercise 5.3 | Q 74.2 | Page 47

A trust fund has Rs 30000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide Rs 30000 among the two types of bonds. If the trust fund must obtain an annual total interest of(ii) Rs 2000

Exercise 5.3 | Q 75 | Page 47

To promote making of toilets for women, an organisation tried to generate awarness through (i) house calls, (ii) letters, and (iii) announcements. The cost for each mode per attempt is given below:

(i) ₹50       (ii) ₹20       (iii) ₹40

The number of attempts made in three villages XY and Z are given below:

(i)               (ii)              (iii)
X      400              300             100
Y      300              250               75
Z      500              400             150

Find the total cost incurred by the organisation for three villages separately, using matrices.

Exercise 5.3 | Q 76 | Page 47

There are 2 families A and B. There are 4 men, 6 women and 2 children in family A, and 2 men, 2 women and 4 children in family B. The recommend daily amount of calories is 2400 for men, 1900 for women, 1800 for children and 45 grams of proteins for men, 55 grams for women and 33 grams for children. Represent the above information using matrix. Using matrix multiplication, calculate the total requirement of calories and proteins for each of the two families. What awareness can you create among people about the planned diet from this question?

Exercise 5.3 | Q 77 | Page 47

In a parliament election, a political party hired a public relations firm to promote its candidates in three ways − telephone, house calls and letters. The cost per contact (in paisa) is given in matrix A as
$A = \begin{bmatrix}140 \\ 200 \\ 150\end{bmatrix}\begin{array} \text{Telephone}\\{\text{House calls }}\\ \text{Letters}\end{array}$

The number of contacts of each type made in two cities X and Y is given in the matrix B as

$\begin{array}"Telephone & House calls & Letters\end{array}$

$B = \begin{bmatrix}1000 & 500 & 5000 \\ 3000 & 1000 & 10000\end{bmatrix}\begin{array} \\City X \\ City Y\end{array}$

Find the total amount spent by the party in the two cities.

What should one consider before casting his/her vote − party's promotional activity of their social activities?

Exercise 5.3 | Q 78 | Page 48

The monthly incomes of Aryan and Babban are in the ratio 3 : 4 and their monthly expenditures are in the ratio 5 : 7. If each saves ₹ 15,000 per month, find their monthly incomes using matrix method. This problem reflects which value?

Exercise 5.3 | Q 79 | Page 48

A trust invested some money in two type of bonds. The first bond pays 10% interest and second bond pays 12% interest. The trust received ₹ 2800 as interest. However, if trust had interchanged money in bonds, they would have got ₹ 100 less as interes. Using matrix method, find the amount invested by the trust.

Exercise 5.4 [Pages 54 - 55]

RD Sharma solutions for Class 12 Maths Chapter 5 Algebra of MatricesExercise 5.4 [Pages 54 - 55]

Exercise 5.4 | Q 1.1 | Page 54

Let  A =[[2,-3],[-7,5]] And B=[[1,0],[2,-4]] verify that

(2A)T = 2AT

Exercise 5.4 | Q 1.2 | Page 54

Let  A =[[2,-3],[-7,5]] And B=[[1,0],[2,-4]] verify that

(A + B)T = AT BT

Exercise 5.4 | Q 1.3 | Page 54

Let  A =[[2,-3],[-7,5]] And B=[[1,0],[2,-4]] verify that

(A − B)T = AT − BT

Exercise 5.4 | Q 1.4 | Page 54

Let  A =[[2,-3],[-7,5]] And B=[[1,0],[2,-4]] verify that

(AB)T = BT AT

Exercise 5.4 | Q 2 | Page 54

If A= [,,] And B=[1  0   4] , Verify that (AB)^T=B^TA^T

Exercise 5.4 | Q 3.1 | Page 54

Let A= [[1,-1,0],[2,1,3],[1,2,1]] And B=[[1,2,3],[2,1,3],[0,1,1]] Find A^T,B^T and verify that   (A + B)T = AT + BT

Exercise 5.4 | Q 3.2 | Page 54

A = \begin{bmatrix}1 & - 1 & 0 \\ 2 & 1 & 3 \\ 1 & 2 & 1\end{bmatrix} and B = \begin{bmatrix}1 & 2 & 3 \\ 2 & 1 & 3 \\ 0 & 1 & 1\end{bmatrix} . Find ATBT and verify that ,

(A B)T = BT + AT

Exercise 5.4 | Q 3.3 | Page 54

Let A= [[1,-1,0],[2,1,3],[1,2,1]] And B=[[1,2,3],[2,1,3],[0,1,1]] Find A^T,B^T and verify that (2A)T = 2AT.

Exercise 5.4 | Q 4 | Page 54

If A=[[-2],,] , B = [1 3 −6], verify that (AB)T = BT AT

Exercise 5.4 | Q 5 | Page 55
If $A = \begin{bmatrix}2 & 4 & - 1 \\ - 1 & 0 & 2\end{bmatrix}, B = \begin{bmatrix}3 & 4 \\ - 1 & 2 \\ 2 & 1\end{bmatrix}$,find (AB)^T

Exercise 5.4 | Q 6.1 | Page 55
For two matrices A and B,   $A = \begin{bmatrix}2 & 1 & 3 \\ 4 & 1 & 0\end{bmatrix}, B = \begin{bmatrix}1 & - 1 \\ 0 & 2 \\ 5 & 0\end{bmatrix}$(AB)T = BT AT.

Exercise 5.4 | Q 6.2 | Page 55
For the matrices A and B, verify that (AB)T = BT AT, where
$A = \begin{bmatrix}1 & 3 \\ 2 & 4\end{bmatrix}, B = \begin{bmatrix}1 & 4 \\ 2 & 5\end{bmatrix}$
Exercise 5.4 | Q 7 | Page 55
If $A^T = \begin{bmatrix}3 & 4 \\ - 1 & 2 \\ 0 & 1\end{bmatrix} and B = \begin{bmatrix}- 1 & 2 & 1 \\ 1 & 2 & 3\end{bmatrix}$ , find AT − BT.

Exercise 5.4 | Q 8 | Page 55

If$A = \begin{bmatrix}\cos \alpha & \sin \alpha \\ - \sin \alpha & \cos \alpha\end{bmatrix}$ , then verify that AT A = I2.

Exercise 5.4 | Q 9 | Page 55
If $A = \begin{bmatrix}\sin \alpha & \cos \alpha \\ - \cos \alpha & \sin \alpha\end{bmatrix}$ , verify that AT A = I2.

Exercise 5.4 | Q 10 | Page 55

If liminii = 1, 2, 3 denote the direction cosines of three mutually perpendicular vectors in space, prove that AAT = I, where $A = \begin{bmatrix}l_1 & m_1 & n_1 \\ l_2 & m_2 & n_2 \\ l_3 & m_3 & n_3\end{bmatrix}$

Exercise 5.5 [Pages 60 - 61]

RD Sharma solutions for Class 12 Maths Chapter 5 Algebra of MatricesExercise 5.5 [Pages 60 - 61]

Exercise 5.5 | Q 1 | Page 60

If$A = \begin{bmatrix}2 & 3 \\ 4 & 5\end{bmatrix}$prove that A − AT is a skew-symmetric matrix.

Exercise 5.5 | Q 2 | Page 61
If $A = \begin{bmatrix}3 & - 4 \\ 1 & - 1\end{bmatrix}$ , show that A − AT is a skewsymmetric matrix.

Exercise 5.5 | Q 3 | Page 61
If the matrix $A = \begin{bmatrix}5 & 2 & x \\ y & z & - 3 \\ 4 & t & - 7\end{bmatrix}$  is a symmetric matrix, find xyz and t.

Exercise 5.5 | Q 4 | Page 61
Let  $A = \begin{bmatrix}3 & 2 & 7 \\ 1 & 4 & 3 \\ - 2 & 5 & 8\end{bmatrix} .$ Find matrices X and Y such that X + Y = A, where X is a symmetric and Y is a skew-symmetric matrix

Exercise 5.5 | Q 5 | Page 61
Express the matrix $A = \begin{bmatrix}4 & 2 & - 1 \\ 3 & 5 & 7 \\ 1 & - 2 & 1\end{bmatrix}$ as the sum of a symmetric and a skew-symmetric matrix.
Exercise 5.5 | Q 6 | Page 61
Define a symmetric matrix. Prove that for
$A = \begin{bmatrix}2 & 4 \\ 5 & 6\end{bmatrix}$, A + AT is a symmetric matrix where AT is the transpose of A.

Exercise 5.5 | Q 7 | Page 61

Express the matrix $A = \begin{bmatrix}3 & - 4 \\ 1 & - 1\end{bmatrix}$  as the sum of a symmetric and a skew-symmetric matrix.

Exercise 5.5 | Q 8 | Page 61

Express the following matrix as the sum of a symmetric and skew-symmetric matrix and verify your result:

$\begin{bmatrix}3 & - 2 & - 4 \\ 3 & - 2 & - 5 \\ - 1 & 1 & 2\end{bmatrix}$

Exercise 5.6 [Pages 62 - 65]

RD Sharma solutions for Class 12 Maths Chapter 5 Algebra of MatricesExercise 5.6 [Pages 62 - 65]

Exercise 5.6 | Q 1 | Page 62

If A is an m × n matrix and B is n × p matrix does AB exist? If yes, write its order.

Exercise 5.6 | Q 2 | Page 62
If  $A = \begin{bmatrix}2 & 1 & 4 \\ 4 & 1 & 5\end{bmatrix}and B = \begin{bmatrix}3 & - 1 \\ 2 & 2 \\ 1 & 3\end{bmatrix}$ . Write the orders of AB and BA.

Exercise 5.6 | Q 3 | Page 62
If $A = \begin{bmatrix}4 & 3 \\ 1 & 2\end{bmatrix} and B = \binom{ - 4}{ 3}$

write AB.

Exercise 5.6 | Q 4 | Page 62

If  $A = \begin{bmatrix}1 \\ 2 \\ 3\end{bmatrix}$ write AAT.

Exercise 5.6 | Q 5 | Page 62

Given an example of two non-zero 2 × 2 matrices A and such that AB = O.

Exercise 5.6 | Q 6 | Page 62
If  $A = \begin{bmatrix}2 & 3 \\ 5 & 7\end{bmatrix}$ , find A + AT.

Exercise 5.6 | Q 7 | Page 62
If  $A = \begin{bmatrix}i & 0 \\ 0 & i\end{bmatrix}$ , write A2.

Exercise 5.6 | Q 8 | Page 62

If $A = \begin{bmatrix}\cos x & \sin x \\ - \sin x & \cos x\end{bmatrix}$ , find x satisfying 0 < x < $\frac{\pi}{2}$ when A + AT = I

Exercise 5.6 | Q 9 | Page 62

If  $A = \begin{bmatrix}\cos x & - \sin x \\ \sin x & \cos x\end{bmatrix}$  , find AAT

Exercise 5.6 | Q 10 | Page 62
If $\begin{bmatrix}1 & 0 \\ y & 5\end{bmatrix} + 2\begin{bmatrix}x & 0 \\ 1 & - 2\end{bmatrix}$  = I, where I is 2 × 2 unit matrix. Find x and y.

Exercise 5.6 | Q 11 | Page 62
If $A = \begin{bmatrix}1 & - 1 \\ - 1 & 1\end{bmatrix}$, satisfies the matrix equation A2 = kA, write the value of k.

Exercise 5.6 | Q 12 | Page 62

If $A = \begin{bmatrix}1 & 1 \\ 1 & 1\end{bmatrix}$ satisfies A4 = λA, then write the value of λ.

Exercise 5.6 | Q 13 | Page 62
If $A = \begin{bmatrix}- 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - 1\end{bmatrix}$ , find A2.

Exercise 5.6 | Q 14 | Page 62

If  $A = \begin{bmatrix}- 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - 1\end{bmatrix}$ , find A3.

Exercise 5.6 | Q 15 | Page 62

If $A = \begin{bmatrix}- 3 & 0 \\ 0 & - 3\end{bmatrix}$ , find A4.

Exercise 5.6 | Q 16 | Page 62

If  [x        2]  [,] = 2 , find x

Exercise 5.6 | Q 17 | Page 62

If A = [aij] is a 2 × 2 matrix such that aij = i + 2j, write A.

Exercise 5.6 | Q 18 | Page 62

Write matrix A satisfying    A+[[2      3],[-1   4]] =[[3     6],[- 3     8]].

Exercise 5.6 | Q 19 | Page 62

If A = [aij] is a square matrix such that aij = i2 − j2, then write whether A is symmetric or skew-symmetric.

Exercise 5.6 | Q 20 | Page 62

For any square matrix write whether AAT is symmetric or skew-symmetric.

Exercise 5.6 | Q 21 | Page 62

If    A =[a_ij]    is a skew-symmetric matrix, then write the value of  Σ_i  a_ij

Exercise 5.6 | Q 22 | Page 62

If A = [aij] is a skew-symmetric matrix, then write the value of  $\sum_i \sum_j$  aij.

Exercise 5.6 | Q 23 | Page 62

If A and B are symmetric matrices, then write the condition for which AB is also symmetric.

Exercise 5.6 | Q 24 | Page 62

If B is a skew-symmetric matrix, write whether the matrix AB AT is symmetric or skew-symmetric.

Exercise 5.6 | Q 25 | Page 63

If B is a symmetric matrix, write whether the matrix AB AT is symmetric or skew-symmetric.

Exercise 5.6 | Q 26 | Page 63

If A is a skew-symmetric and n ∈ N such that (An)T = λAn, write the value of λ.

Exercise 5.6 | Q 27 | Page 63

If A is a symmetric matrix and n ∈ N, write whether An is symmetric or skew-symmetric or neither of these two.

Exercise 5.6 | Q 28 | Page 63

If A is a skew-symmetric matrix and n is an even natural number, write whether An is symmetric or skew symmetric or neither of these two.

Exercise 5.6 | Q 29 | Page 63

If A is a skew-symmetric matrix and n is an odd natural number, write whether An is symmetric or skew-symmetric or neither of the two.

Exercise 5.6 | Q 30 | Page 63

If A and B are symmetric matrices of the same order, write whether AB − BA is symmetric or skew-symmetric or neither of the two.

Exercise 5.6 | Q 31 | Page 63

Write a square matrix which is both symmetric as well as skew-symmetric.

Exercise 5.6 | Q 32 | Page 63

Find the values of x and y, if $2\begin{bmatrix}1 & 3 \\ 0 & x\end{bmatrix} + \begin{bmatrix}y & 0 \\ 1 & 2\end{bmatrix} = \begin{bmatrix}5 & 6 \\ 1 & 8\end{bmatrix}$

Exercise 5.6 | Q 33 | Page 63

If  $\begin{bmatrix}x + 3 & 4 \\ y - 4 & x + y\end{bmatrix} = \begin{bmatrix}5 & 4 \\ 3 & 9\end{bmatrix}$ , find x and y

Exercise 5.6 | Q 34 | Page 63

Find the value of x from the following: [[2x - y          5],[ 3                         y ]] = [[6            5 ],[3         - 2\]]

Exercise 5.6 | Q 35 | Page 63

Find the value of y, if $\begin{bmatrix}x - y & 2 \\ x & 5\end{bmatrix} = \begin{bmatrix}2 & 2 \\ 3 & 5\end{bmatrix}$

Exercise 5.6 | Q 36 | Page 63

Find the value of x, if  $\begin{bmatrix}3x + y & - y \\ 2y - x & 3\end{bmatrix} = \begin{bmatrix}1 & 2 \\ - 5 & 3\end{bmatrix}$

Exercise 5.6 | Q 37 | Page 63

If matrix A = [1 2 3], write AAT.

Exercise 5.6 | Q 38 | Page 63

if  $\begin{bmatrix}2x + y & 3y \\ 0 & 4\end{bmatrix} = \begin{bmatrix}6 & 0 \\ 6 & 4\end{bmatrix}$  , then find x.

Exercise 5.6 | Q 39 | Page 63

If $A = \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}$ , find A + AT.

Exercise 5.6 | Q 40 | Page 63

If $\begin{bmatrix}a + b & 2 \\ 5 & b\end{bmatrix} = \begin{bmatrix}6 & 5 \\ 2 & 2\end{bmatrix}$ , then find a.

Exercise 5.6 | Q 41 | Page 63

If A is a matrix of order 3 × 4 and B is a matrix of order 4 × 3, find the order of the matrix of AB

Exercise 5.6 | Q 42 | Page 63

If $A = \begin{bmatrix}\cos \alpha & - \sin \alpha \\ \sin \alpha & \cos \alpha\end{bmatrix}$ is identity matrix, then write the value of α.

Exercise 5.6 | Q 43 | Page 63

If  $\begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}\begin{bmatrix}3 & 1 \\ 2 & 5\end{bmatrix} = \begin{bmatrix}7 & 11 \\ k & 23\end{bmatrix}$ ,then write the value of k.

Exercise 5.6 | Q 44 | Page 63

If I is the identity matrix and A is a square matrix such that A2 = A, then what is the value of (I + A)2 = 3A?

Exercise 5.6 | Q 45 | Page 63

If $A = \begin{bmatrix}1 & 2 \\ 0 & 3\end{bmatrix}$ is written as B + C, where B is a symmetric matrix and C is a skew-symmetric matrix, then B is equal to.

Exercise 5.6 | Q 46 | Page 63

If A is 2 × 3 matrix and B is a matrix such that AT B and BAT both are defined, then what is the order of B ?

Exercise 5.6 | Q 47 | Page 64

What is the total number of 2 × 2 matrices with each entry 0 or 1?

Exercise 5.6 | Q 48 | Page 64

If $\begin{bmatrix}x & x - y \\ 2x + y & 7\end{bmatrix} = \begin{bmatrix}3 & 1 \\ 8 & 7\end{bmatrix}$  , then find the value of y.

Exercise 5.6 | Q 49 | Page 64

If a matrix has 5 elements, write all possible orders it can have.

Exercise 5.6 | Q 50 | Page 64

For a 2 × 2 matrix A = [aij] whose elements are given by

$a_{ij} = \frac{i}{j}$ , write the value of a12.

Exercise 5.6 | Q 51 | Page 64

If  $x\binom{2}{3} + y\binom{ - 1}{1} = \binom{10}{5}$ , find the value of x.

Exercise 5.6 | Q 52 | Page 64

If  $\begin{bmatrix}9 & - 1 & 4 \\ - 2 & 1 & 3\end{bmatrix} = A + \begin{bmatrix}1 & 2 & - 1 \\ 0 & 4 & 9\end{bmatrix}$ , then find matrix A.

Exercise 5.6 | Q 53 | Page 64

If  $\begin{bmatrix}a - b & 2a + c \\ 2a - b & 3c + d\end{bmatrix} = \begin{bmatrix}- 1 & 5 \\ 0 & 13\end{bmatrix}$ , find the value of b.

Exercise 5.6 | Q 54 | Page 64

For what value of x, is the matrix  $A = \begin{bmatrix}0 & 1 & - 2 \\ - 1 & 0 & 3 \\ x & - 3 & 0\end{bmatrix}$  a skew-symmetric matrix?

Exercise 5.6 | Q 55 | Page 64

If matrix  $A = \begin{bmatrix}2 & - 2 \\ - 2 & 2\end{bmatrix}$  and A2 = pA, then write the value of p.

Exercise 5.6 | Q 56 | Page 64

If A is a square matrix such that A2 = A, then write the value of 7A − (I + A)3, where I is the identity matrix.

Exercise 5.6 | Q 57 | Page 64

If  $2\begin{bmatrix}3 & 4 \\ 5 & x\end{bmatrix} + \begin{bmatrix}1 & y \\ 0 & 1\end{bmatrix} = \begin{bmatrix}7 & 0 \\ 10 & 5\end{bmatrix}$ , find x − y.

Exercise 5.6 | Q 58 | Page 64

If $\begin{bmatrix}x & 1\end{bmatrix}\begin{bmatrix}1 & 0 \\ - 2 & 0\end{bmatrix} = O$  , find x.

Exercise 5.6 | Q 59 | Page 64

If $\begin{bmatrix}a + 4 & 3b \\ 8 & - 6\end{bmatrix} = \begin{bmatrix}2a + 2 & b + 2 \\ 8 & a - 8b\end{bmatrix}$ , write the value of a − 2b.

Exercise 5.6 | Q 60 | Page 64

Write a 2 × 2 matrix which is both symmetric and skew-symmetric.

Exercise 5.6 | Q 61 | Page 64

If  $\begin{bmatrix}xy & 4 \\ z + 6 & x + y\end{bmatrix} = \begin{bmatrix}8 & w \\ 0 & 6\end{bmatrix}$ , write the value of (x + y + z).

Exercise 5.6 | Q 62 | Page 64

Construct a 2 × 2 matrix A = [aij] whose elements aij are given by $a_{ij} = \begin{cases}\frac{\left| - 3i + j \right|}{2} & , if i \neq j \\ \left( i + j \right)^2 & , if i = j\end{cases}$

Exercise 5.6 | Q 63 | Page 64

If  $\binom{x + y}{x - y} = \begin{bmatrix}2 & 1 \\ 4 & 3\end{bmatrix}\binom{1}{ - 2}$ , then write the value of (xy).

Exercise 5.6 | Q 64 | Page 64

Matrix A = $\begin{bmatrix}0 & 2b & - 2 \\ 3 & 1 & 3 \\ 3a & 3 & - 1\end{bmatrix}$  is given to be symmetric, find values of a and b.

Exercise 5.6 | Q 65 | Page 65

Write the number of all possible matrices of order 2 × 2 with each entry 1, 2 or 3.

Exercise 5.6 | Q 66 | Page 65

If [2     1       3]([-1,0,-1],[-1,1,0],[0,1,1])(,,[-1])=A , then write the order of matrix A.

Exercise 5.6 | Q 67 | Page 65

If A = ([3   5] , [7     9]) is written as A = P + Q, where as A = p + Q , Where  P is a symmetric matrix and Q is skew symmetric matrix , then wqrite the matrix P.

Exercise 5.6 | Q 68 | Page 65

Let and be matrices of orders 3 x 2 and 2 x

4 respectively. Write the order of matrix AB.

Exercise 5.7 [Pages 38 - 69]

RD Sharma solutions for Class 12 Maths Chapter 5 Algebra of MatricesExercise 5.7 [Pages 38 - 69]

Exercise 5.7 | Q 1 | Page 65

If $A = \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & - 1\end{bmatrix}$ , then A2 is equal to ___________ .

• a null matrix

• a unit matrix

• A

Exercise 5.7 | Q 2 | Page 66

If A=[[i,0],[0,i ]] , n ∈ N, then A4n equals

• [[0,I],[I,0]]

• [[0,0],[0,0]]

• [[1,0],[0,1]]

• [[0,I],[I,0]]

Exercise 5.7 | Q 3 | Page 66

If A and B are two matrices such that AB = A and BA = B, then B2 is equal to

• B

• A

• 1

• 0

Exercise 5.7 | Q 4 | Page 66

If AB = A and BA = B, where A and B are square matrices,  then

• B2 = B and A2 = A

• B2B and A2 = A

• A2 A , B2 =B

• A2 A , B2 ≠ B

Exercise 5.7 | Q 5 | Page 66

If A and B are two matrices such n  that AB = B and BA = A , A^2 + B^2 is equal to

• AB

• BA

• A +

•  AB

Exercise 5.7 | Q 6 | Page 66

If  $\begin{bmatrix}\cos\frac{2\pi}{7} & - \sin\frac{2\pi}{7} \\ \sin\frac{2\pi}{7} & \cos\frac{2\pi}{7}\end{bmatrix}^k = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$ then the least positive integral value of k is _____________.

• 3

• 4

• 6

• 7

Exercise 5.7 | Q 7 | Page 66

If the matrix AB is zero, then

• It is not necessary that either A = O or, B = O

• A = O or B = O

• A = O and B = O

• all the above statements are wrong

Exercise 5.7 | Q 8 | Page 66

Let A = $\begin{bmatrix}a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a\end{bmatrix}$, then An is equal to

• \begin{bmatrix}a^n & 0 & 0 \\ 0 & a^n & 0 \\ 0 & 0 & a\end{bmatrix}

• $\begin{bmatrix}a^n & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a\end{bmatrix}$

• $\begin{bmatrix}a^n & 0 & 0 \\ 0 & a^n & 0 \\ 0 & 0 & a^n\end{bmatrix}$

•  $\begin{bmatrix}na & 0 & 0 \\ 0 & na & 0 \\ 0 & 0 & na\end{bmatrix}$

Exercise 5.7 | Q 9 | Page 66

If AB are square matrices of order 3, A is non-singular and AB = O, then B is a

• null matrix

• singular matrix

• unit-matrix

• non-singular matrix

Exercise 5.7 | Q 10 | Page 66

If $A = \begin{bmatrix}n & 0 & 0 \\ 0 & n & 0 \\ 0 & 0 & n\end{bmatrix} \text {and B} = \begin{bmatrix}a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c^1 & c_2 & c_3\end{bmatrix}$then AB is equal to

• nB

• B^n

• A+B

Exercise 5.7 | Q 11 | Page 66

If  $A = \begin{bmatrix}1 & a \\ 0 & 1\end{bmatrix}$then An (where n ∈ N) equals

•  $\begin{bmatrix}1 & na \\ 0 & 1\end{bmatrix}$

•  $\begin{bmatrix}1 & n^2 a \\ 0 & 1\end{bmatrix}$

• $\begin{bmatrix}1 & na \\ 0 & 0\end{bmatrix}$

•  $\begin{bmatrix}n & na \\ 0 & n\end{bmatrix}$

Exercise 5.7 | Q 12 | Page 66

If  $A = \begin{bmatrix}1 & 2 & x \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix} and B = \begin{bmatrix}1 & - 2 & y \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$ and AB = I3, then x + y equals

• 0

• 1

• 2

• none of these

Exercise 5.7 | Q 13 | Page 66

If $A = \begin{bmatrix}1 & - 1 \\ 2 & - 1\end{bmatrix}, B = \begin{bmatrix}a & 1 \\ b & - 1\end{bmatrix}$and (A + B)2 = A2 + B2,   values of a and b are

• a = 4, b = 1

• a = 1, b = 4

• a = 0, b = 4

•  a = 2, b = 4

Exercise 5.7 | Q 14 | Page 67

If  $A = \begin{bmatrix}\alpha & \beta \\ \gamma & - \alpha\end{bmatrix}$  is such that A2 = I, then

• 1 + α2 + βγ = 0

• 1 − α2 + βγ = 0

• 1 − α2 − βγ = 0

• 1 + α2 − βγ = 0

Exercise 5.7 | Q 15 | Page 67

If S = [Sij] is a scalar matrix such that sij = k and A is a square matrix of the same order, then AS = SA = ?

• Ak

•  k +

• kA

• kS

Exercise 5.7 | Q 16 | Page 67

If A is a square matrix such that A2 = A, then (I + A)3 − 7A is equal to

• A

• I-A

• I

• 3A

Exercise 5.7 | Q 17 | Page 67

If a matrix A is both symmetric and skew-symmetric, then

• A is a diagonal matrix

•  A is a zero matrix

•  A is a scalar matrix

• A is a square matrix

Exercise 5.7 | Q 18 | Page 67

The matrix $\begin{bmatrix}0 & 5 & - 7 \\ - 5 & 0 & 11 \\ 7 & - 11 & 0\end{bmatrix}$ is

•  a skew-symmetric matrix

• a symmetric matrix

• a diagonal matrix

• an uppertriangular matrix

Exercise 5.7 | Q 19 | Page 67

If A is a square matrix, then AA is a

• skew-symmetric matrix

• symmetric matrix

• diagonal matrix

• none of these

Exercise 5.7 | Q 20 | Page 67

If A and B are symmetric matrices, then ABA is

• symmetric matrix

• skew-symmetric matrix

• diagonal matrix

• scalar matrix

Exercise 5.7 | Q 21 | Page 67

If $A = \begin{bmatrix}5 & x \\ y & 0\end{bmatrix}$  and A = AT, then

• x = 0, y = 5

•  x + y = 5

•  x =

• none of these

Exercise 5.7 | Q 22 | Page 67

If A is 3 × 4 matrix and B is a matrix such that A'B and BA' are both defined. Then, B is of the type

• 3 × 4

• 3 × 3

• 4 × 4

• 4 × 3

Exercise 5.7 | Q 23 | Page 67

If A = [aij] is a square matrix of even order such that aij = i2 − j2, then

• A is a skew-symmetric matrix and  | A | = 0

•  A is symmetric matrix and | A | is a square

•  A is symmetric matrix and | A | = 0

• none of these.

Exercise 5.7 | Q 24 | Page 67

If $A = \begin{bmatrix}\cos \theta & - \sin \theta \\ \sin \theta & \cos \theta\end{bmatrix}$  then AT + A = I2, if

• θ = n π, n ∈ Z

• θ     = (2n + 1) $\frac{\pi}{2}$ n ∈

• θ = 2n π +$\frac{\pi}{3}$ n ∈ Z

• none of these

Exercise 5.7 | Q 25 | Page 67

If $A = \begin{bmatrix}2 & 0 & - 3 \\ 4 & 3 & 1 \\ - 5 & 7 & 2\end{bmatrix}$  is expressed as the sum of a symmetric and skew-symmetric matrix, then the symmetric matrix is

• $\begin{bmatrix}2 & 2 & - 4 \\ 2 & 3 & 4 \\ - 4 & 4 & 2\end{bmatrix}$

•  $\begin{bmatrix}2 & 4 & - 5 \\ 0 & 3 & 7 \\ - 3 & 1 & 2\end{bmatrix}$

• $\begin{bmatrix}4 & 4 & - 8 \\ 4 & 6 & 8 \\ - 8 & 8 & 4\end{bmatrix}$

• $\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$

Exercise 5.7 | Q 26 | Page 68

Out of the given matrices, choose that matrix which is a scalar matrix:

• $\begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}$

•  $\begin{bmatrix}0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}$

•  $\begin{bmatrix}0 & 0 \\ 0 & 0 \\ 0 & 0\end{bmatrix}$

• $\begin{bmatrix}0 \\ 0 \\ 0\end{bmatrix}$

Exercise 5.7 | Q 27 | Page 68

The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is

• 27

• 18

• 81

• 512

Exercise 5.7 | Q 28 | Page 68

Which of the given values of x and y make the following pairs of matrices equal? $\begin{bmatrix}3x + 7 & 5 \\ y + 1 & 2 - 3x\end{bmatrix}, \begin{bmatrix}0 & y - 2 \\ 8 & 4\end{bmatrix}$

• x =$- \frac{1}{3}$,y = 7

•  y = 7, x = $- \frac{2}{3}$

• x =  $- \frac{1}{3}$ 4 =$- \frac{2}{5}$

• Not possible to find

Exercise 5.7 | Q 29 | Page 68

If $A = \begin{bmatrix}0 & 2 \\ 3 & - 4\end{bmatrix}$  and $kA = \begin{bmatrix}0 & 3a \\ 2b & 24\end{bmatrix}$  then the values of kab, are respectively

•  −6, −12, −18

•  −6, 4, 9

• −6, −4, −9

• −6, 12, 18

Exercise 5.7 | Q 30 | Page 68

If $I = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}, J = \begin{bmatrix}0 & 1 \\ - 1 & 0\end{bmatrix} and B = \begin{bmatrix}\cos \theta & \sin \theta \\ - \sin \theta & \cos \theta\end{bmatrix}$ then B equals )

• I cos θ + J sin θ

• I sin θ + J cos θ

• I cos θ − J sin θ

• I cos θ + J sin θ

Exercise 5.7 | Q 31 | Page 68

The trace of the matrix $A = \begin{bmatrix}1 & - 5 & 7 \\ 0 & 7 & 9 \\ 11 & 8 & 9\end{bmatrix}$, is

• 17

• 25

• 3

• 12

Exercise 5.7 | Q 32 | Page 68

If A = [aij] is a scalar matrix of order n × n such that aii = k, for all i, then trace of A is equal to

• nk

• n +

• $\frac{n}{k}$

• none of these

Exercise 5.7 | Q 33 | Page 68

The matrix  $A = \begin{bmatrix}0 & 0 & 4 \\ 0 & 4 & 0 \\ 4 & 0 & 0\end{bmatrix}$ is a

• square matrix

• diagonal matrix

• unit matrixn

• none of these

Exercise 5.7 | Q 34 | Page 68

The number of possible matrices of order 3 × 3 with each entry 2 or 0 is

• 9

• 27

• 81

• none of these

Exercise 5.7 | Q 35 | Page 68

If $\begin{bmatrix}2x + y & 4x \\ 5x - 7 & 4x\end{bmatrix} = \begin{bmatrix}7 & 7y - 13 \\ y & x + 6\end{bmatrix}$

• x = 3 , y =-1

• x = 2 , y= 3

• x= 2 , y= 4

• x = 3, y= 3

Exercise 5.7 | Q 36 | Page 38

If A is a square matrix such that A2 = I, then (A − I)3 + (A + I)3 − 7A is equal to

• A

• I-A

• I+A

• 3A

Exercise 5.7 | Q 37 | Page 69

If A and B are two matrices of order 3 × m and 3 × n respectively and m = n, then the order of 5A − 2B is

• × 3

• 3 × 3

• m ×

• 3 × n

Exercise 5.7 | Q 38 | Page 69

If A is a matrix of order m × n and B is a matrix such that ABT and BTA are both defined, then the order of matrix B is

Disclaimer: option (a) and (d) both are the same.

•  m × n

• n  × n

• n × m

• m  × n

Exercise 5.7 | Q 39 | Page 69

If A and B are matrices of the same order, then ABT − BAT is a

•  skew symmetric matrix

• null matrix

• unit matrix

• symmetric matrix

Exercise 5.7 | Q 40 | Page 69

If matrix  $A = \left[ a_{ij} \right]_{2 \times 2}$ where

$a_{ij} = \begin{cases}1 & , if i \neq j \\ 0 & , if i = j\end{cases}$

• I

• A

• O

• -I

Exercise 5.7 | Q 41 | Page 69

If $A = \frac{1}{\pi}\begin{bmatrix}\sin^{- 1} \left( \ pix \right) & \ tan^{- 1} \left( \frac{x}{\pi} \right) \\ \sin^{- 1} \left( \frac{x}{\pi} \right) & \cot^{- 1} \left( \ pix \right)\end{bmatrix}, B = \frac{1}{\pi}\begin{bmatrix}- \cos^{- 1} \left( \ pix \right) & \tan^{- 1} \left( \frac{x}{\pi} \right) \\ \sin^{- 1} \left( \frac{x}{\pi} \right) & - \tan^{- 1} \left( \ pix \right)\end{bmatrix}$

A − B is equal to

• I

• 0

• 2I

• 1/2 I

Exercise 5.7 | Q 42 | Page 69

If A and B are square matrices of the same order, then (A + B)(A − B) is equal to

•  A2 − B2

•  A2 − BA − AB − B2

• A2 − B2 + BA − AB

• A2  BA + B+ AB

Exercise 5.7 | Q 43 | Page 69

If  $A = \begin{bmatrix}2 & - 1 & 3 \\ - 4 & 5 & 1\end{bmatrix}\text{ and B }= \begin{bmatrix}2 & 3 \\ 4 & - 2 \\ 1 & 5\end{bmatrix}$ then

• only AB is defined

• only BA is defined

• AB and BA both are defined

• AB and BA both are not defined

Exercise 5.7 | Q 44 | Page 69

The matrix  $A = \begin{bmatrix}0 & - 5 & 8 \\ 5 & 0 & 12 \\ - 8 & - 12 & 0\end{bmatrix}$ is a

• diagonal matrix

• symmetric matrix

• skew-symmetric matrix

• scalar matrix

Exercise 5.7 | Q 45 | Page 69

The matrix   $A = \begin{bmatrix}1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 4\end{bmatrix}$ is

• identity matrix

• symmetric matrix

• skew-symmetric matrix

• diagonal matrix

Chapter 5: Algebra of Matrices

Exercise 5.1Exercise 5.2Exercise 5.3Exercise 5.4Exercise 5.5Exercise 5.6Exercise 5.7 RD Sharma solutions for Class 12 Maths chapter 5 - Algebra of Matrices

RD Sharma solutions for Class 12 Maths chapter 5 (Algebra of Matrices) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Class 12 Maths solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 12 Maths chapter 5 Algebra of Matrices are Introduction of Operations on Matrices, Inverse of a Nonsingular Matrix by Elementary Transformation, Multiplication of Two Matrices, Negative of Matrix, Properties of Matrix Addition, Concept of Transpose of a Matrix, Subtraction of Matrices, Addition of Matrices, Symmetric and Skew Symmetric Matrices, Types of Matrices, Proof of the Uniqueness of Inverse, Invertible Matrices, Multiplication of Matrices, Properties of Multiplication of Matrices, Equality of Matrices, Order of a Matrix, Matrices Notation, Introduction of Matrices, Multiplication of a Matrix by a Scalar, Properties of Scalar Multiplication of a Matrix, Properties of Transpose of the Matrices, Elementary Transformations.

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