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# NCERT solutions Mathematics Class 12 Part 1 chapter 3 Matrices

#### Chapters

NCERT Solutions for Mathematics Class 12 Part 2

## Chapter 3 - Matrices

#### Pages 64 - 80

In the matrix A = [(2,5,19,-7),(35,-2, 5/2 ,12), (sqrt3, 1, -5 , 17)]

The order of the matrix

Q 1.1 | Page 64

Let A = [(2,4),(3,2)] , B = [(1,3),(-2,5)], C = [(-2,5),(3,4)]

Find A - B

Q 1.2 | Page 80

In the matrix A = [(2,5,19,-7),(35,-2, 5/2 ,12), (sqrt3, 1, -5 , 17)]

The number of elements,

Q 1.2 | Page 64

In the matrix A = [(2,5,19,-7),(35,-2, 5/2 ,12), (sqrt3, 1, -5 , 17)] Write the elements a13a21a33a24a23

Q 1.3 | Page 64

If a matrix has 24 elements, what are the possible order it can have? What, if it has 13 elements?

Q 2 | Page 64

If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?

Q 3 | Page 64

Construct a 2 × 2 matrix, A = [a_(ij)], whose elements are given by:

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

Q 4.1 | Page 64

Construct a 2 × 2 matrix, A= [a_(ij)], whose elements are given by a_(ij) = i/j

Q 4.2 | Page 64

Construct a 2 × 2 matrix, A = [a_(ij)]  whose elements are given by:

a_(ij) = (1 + 2j)^2/2

Q 4.3 | Page 64

Construct a 3 × 4 matrix, whose elements are given by a_(ij) = 1/2 |-3i + j|

Q 5.1 | Page 64

Construct a 3 × 4 matrix, whose elements are given by a_(ij) = 2i - j

Q 5.2 | Page 64

Find the value of xy, and z from the following equation:

[(4,3),(x,5)] = [(y,z),(1,5)]

Q 6.1 | Page 64

Find the value of xy, and z from the following equation:

[(x+y, 2),(5+z, xy)] = [(6,2), (5,8)]

Q 6.2 | Page 64

Find the value of xy, and z from the following equation:

[(x+y+z), (x+z), (y+z)] = [(9),(5),(7)]

Q 6.3 | Page 64

Find the value of abc, and d from the equation:

[(a-b, 2a+c),(2a-b, 3x+d)] = [(-1,5),(0,13)]

Q 7 | Page 64

A = [a_(ij)]_(mxxn) is a square matrix, if

(A) m < n

(B) m > n

(C) m = n

(D) None of these

Q 8 | Page 65

Which of the given values of x and y make the following pair of matrices equal

[(3x+7, 5),(y+1, 2-3x)] = [(0,y-2),(8,4)]

(A) x= (-1)/3, y = 7

(B) Not possible to find

(C) y = 7, x = (-2)/3

(D) x = (-1)/3, y = (-2)/3

Q 9 | Page 65

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

(A) 27

(B) 18

(C) 81

(D) 512

Q 10 | Page 65

#### Pages 80 - 83

Let A = [(2,4),(3,2)] , B = [(1,3),(-2,5)], C = [(-2,5),(3,4)]

Find  A + B

Q 1.1 | Page 80

Let A = [(2,4),(3,2)] , B = [(1,3),(-2,5)], C = [(-2,5),(3,4)]

Find  3A -C

Q 1.3 | Page 80

Let A = [(2,4),(3,2)] , B = [(1,3),(-2,5)], C = [(-2,5),(3,4)]   Find AB

Q 1.4 | Page 80

Let A = [(2,4),(3,2)] , B = [(1,3),(-2,5)], C = [(-2,5),(3,4)]

Find BA

Q 1.5 | Page 80

Compute the following: [(a,b),(-b, a)] + [(a,b),(b,a)]

Q 2.1 | Page 80

Compute the following:

[(a^2+b^2, b^2+c^2),(a^2+c^2, a^2+b^2)] + [(2ab , 2bc),(-2ac, -2ab)]

Q 2.2 | Page 80

Compute the following:

[(-1,4, -6),(8,5,16),(2,8,5)] + [(12,7,6),(8,0,5),(3,2,4)]

Q 2.3 | Page 80

Compute the following:

[(cos^2x, sin^2 x),(sin^2 x ,cos^2 x)]+[(sin^2 x, cos^2 x), (cos^2 x, sin^2 x)]

Q 2.5 | Page 80

Compute the indicated products

[(a,b),(-b,a)][(a,-b),(b,a)]

Q 3.1 | Page 80

Compute the indicated products

[(1),(2),(3)] [2,3,4]

Q 3.2 | Page 80

Compute the indicated products

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

Q 3.3 | Page 80

Compute the indicated products

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

Q 3.4 | Page 80

Compute the indicated products

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

Q 3.5 | Page 80

Compute the indicated products

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

Q 3.6 | Page 80

if A = [(1,2,-3),(5,0,2),(1,-1,1)], B = [(3,-1,2),(4,2,5),(2,0,3)] and C = [(4,1,2),(0,3,2),(1,-2,3)] then compute (A + B) and (B - C). Also verify that A + (B -C) = (A +B) - C

Q 4 | Page 81

if  A = [(2/3, 1, 5/3), (1/3, 2/3, 4/3),(7/3, 2, 2/3)] and B = [(2/5, 3/5,1),(1/5, 2/5, 4/5), (7/5,6/5, 2/5)] then compute 3A - 5B.

Q 5 | Page 81

Simplify  cos theta[(cos theta, sintheta),(-sin theta, cos theta)] + sin theta [(sin theta, -cos theta), (cos theta, sin theta)]

Q 6 | Page 81

Find X and Y, if X + Y = [(7,0),(2,5)] and X - Y = [(3,0),(0,3)]

Q 7.1 | Page 81

Find and Y, if 2X + 3Y = [(2,3),(4,0)] and 3X + 2Y = [(2, -1),(-1,5)]

Q 7.2 | Page 81

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

Q 8 | Page 81

Find x and y, if  2[(1,3),(0, x)]+[(y,0),(1,2)] = [(5,6),(1,8)]

Q 9 | Page 81

Solve the equation for x, y, z and t if 2[(x,z),(y, t)] + 3[(1,-1),(0,2)] = 3[(3,5),(4,6)]

Q 10 | Page 81

if x[2/3] + y[(-1),(1)] = [10/5], find values of x and y.

Q 11 | Page 81

Given 3[(x,y),(z,w)] = [(x,6),(-1,2W)] + [(4,x+y),(Z+W,3)] find the values of xyz and w

Q 12 | Page 81

If F(x) = [(cosx, -sinx,0),(sinx, cosx, 0),(0,0,1)]  show that F(x)F(y) = F(x + y)

Q 13 | Page 82

Show that [(5, -1),(6,7)][(2,1),(3,4)] != [(2,1),(3,4)][(5,-1),(6,7)]

Q 14.1 | Page 82

Show that [(1,2,3),(0,1,0),(1,1,0)][(-1,1,0),(0,-1,1),(2,3,4)]!=[(-1,1,0),(0,-1,1),(2,3,4)][(1,2,3),(0,1,0),(1,1,0)]

Q 14.2 | Page 82

Find A^2 - 5A + 6I if A = [(2,0,1),(2,1,3),(1,-1,0)]

Q 15 | Page 82

if A = [(1,0,2),(0,2,1),(2,0,3)] , prove that A^2 - 6A^2 + 7A + 2I = 0

Q 16 | Page 82

if A = [(3, -2),(4,-2)] and l = Matric [(1,0),(0,1)]  find k so that A^2 = kA - 2I

Q 17 | Page 82

if A = [(0, -tan alpha/2), (tan alpha/2, 0)] and I is the identity matrix of order 2, show that I + A = (I -A)[(cos alpha, -sin alpha),(sin alpha, cos alpha)]

Q 18 | Page 82

A trust fund has Rs 30,000 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 30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of:

(a) Rs 1,800 (b) Rs 2,000

Q 19 | Page 82

The bookshop of a particular school has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economics books. Their selling prices are Rs 80, Rs 60 and Rs 40 each respectively. Find the total amount the bookshop will receive from selling all the books using matrix algebra.

Q 20 | Page 82

Assume XYZW and P are matrices of order 2 x n, 3 x k, 2 x p,n x 3 and respectively. The restriction on nk and p so that PY + WY will be defined are:

A. k = 3, p = n

B. k is arbitrary, p = 2

C. p is arbitrary, k = 3

D. k = 2, p = 3

Q 21 | Page 83

Assume XYZW and P are matrices of order 2 x n, 3 x k, 2 x p, n x 3, and p x k respectively. If n = p, then the order of the matrix is &X - 5Z

A p × 2 B 2 × n C n × 3 D p × n

Q 22 | Page 83

#### Pages 88 - 90

Find the transpose the matrices [(5),(1/2),(-1)]

Q 1 | Page 88

Find the transpose of matrices [(1,-1),(2,3)]

Q 1.2 | Page 88

Find the transpose of matrices [(-1,5,6),(sqrt3, 5, 6),(2,3,-1)]

Q 1.3 | Page 88

if A = [(-1,2,3),(5,7,9),(-2,1,1)] and B = [(-4,1,-5),(1,2,0),(1,3,1)] then verify that

(A+ B)' = A' + B'

Q 2.1 | Page 88

if A = [(-1,2,3),(5,7,9),(-2,1,1)] and B = [(-4,1,-5),(1,2,0),(1,3,1)] then verify that

(A- B)' = A' - B'

Q 2.2 | Page 88

if A' [(3,4),(-1, 2),(0,1)] and B = [((-1,2,1),(1,2,3))] then verify that (A + B)' = A' + B'

Q 3.1 | Page 88

if A' [(3,4),(-1, 2),(0,1)] and B = [((-1,2,1),(1,2,3))] then verify that (A - B)' = A' - B'

Q 3.2 | Page 88

if A' = [(-2,3),(1,2)] and B = [(-1,0),(1,2)]  then find (A + 2B)'

Q 4 | Page 88

For the matrices A and B, verify that (AB)′ = B'A' where

A =[(1),(-4), (3)], B = [-1, 2 1]

Q 5.1 | Page 88

For the matrices A and B, verify that (AB)′ = B'A'  where

A =[(0), (1),(2)] , B =[1 , 5, 7]

Q 5.2 | Page 88

if A = [(cos alpha, sin alpha), (-sin alpha, cos alpha)] then verify that  A'A = I

Q 6.1 | Page 89

if A = [(sin alpha, cos alpha),(-cos alpha, sin alpha)] then verify that  A'A = I

Q 6.2 | Page 89

Show that the matrix  A = [(1,-1,5),(-1,2,1),(5,1,3)] is a symmetric matrix

Q 7.1 | Page 89

Show that the matrix  A = [(0,1,-1),(-1,0,1),(1,-1,0)] is a skew symmetric matrix

Q 7.2 | Page 89

For the matrix A = [(1,5),(6,7)] verify that (A + A') is a symmetric matrix

Q 8.1 | Page 89

For the matrix A = [(1,5),(6,7)] verify that (A - A') is a skew symmetric matrix

Q 8.2 | Page 89

Find 1/2 (A + A') and 1/2 (A -A') When A = [(0, a, b),(-a,0,c),(-b,-c,0)]

Q 9 | Page 89

Express the matrices as the sum of a symmetric and a skew symmetric matrix:

[(3,5),(1,-1)]

Q 10.1 | Page 89

Express the matrices as the sum of a symmetric and a skew symmetric matrix:

[(6, -2,2),(-2,3,-1),(2,-1,3)]

Q 10.2 | Page 89

Express the matrices as the sum of a symmetric and a skew symmetric matrix:

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

Q 10.3 | Page 89

Express the following matrices as the sum of a symmetric and a skew symmetric matrix:

[(1,5),(-1,2)]

Q 10.4 | Page 89

If AB are symmetric matrices of same order, then AB − BA is a

A. Skew symmetric matrix B. Symmetric matrix

C. Zero matrix D. Identity matrix

Q 11 | Page 90

if A= [(cos alpha, -sin alpha),(sin alpha, cos alpha)] then A + A' = I if the value of α is

A pi/6

B pi/3

C pi

D (3pi)/2

Q 12 | Page 90

#### Pages 97 - 100

Find the inverse of each of the matrices, if it exists. [(1, -1),(2,3)]

Q 1 | Page 97

if A = [(1,1,1),(1,1,1),(1,1,1)], Prove that A" = [(3^(n-1),3^(n-1),3^(n-1)),(3^(n-1),3^(n-1),3^(n-1)),(3^(n-1),3^(n-1),3^(n-1))] n in N

Q 2 | Page 100

Find the inverse of each of the matrices, if it exists. [(2,1),(1,1)]

Q 2 | Page 97

if A = [(3,-4),(1,-1)] then prove A"= [(1+2n, -4n),(n, 1-2n)] where n is any positive integer

Q 3 | Page 100

Find the inverse of each of the matrices, if it exists.

[(1,3),(2,7)]

Q 3 | Page 97

Find the inverse of each of the matrices, if it exists.

[(2,3),(5,7)]

Q 4 | Page 97

Find the inverse of each of the matrices, if it exists.

[(2,7),(1,4)]

Q 5 | Page 97

Find the inverse of each of the matrices, if it exists.

[(2,5),(1,3)]

Q 6 | Page 97

Find the inverse of each of the matrices, if it exists.

[(3,1),(5,2)]

Q 7 | Page 97

Find the inverse of each of the matrices, if it exists.

[(4,5),(3,4)]

Q 8 | Page 97

Find the inverse of each of the matrices, if it exists.

[(3,10),(2,7)]

Q 9 | Page 97

Find the inverse of each of the matrices, if it exists.

[(3,-1),(-4,2)]

Q 10 | Page 97

Find the inverse of each of the matrices, if it exists.

[(2, -6),(1, -2)]

Q 11 | Page 97

Find the inverse of each of the matrices, if it exists.

[(6,-3),(-2,1)]

Q 12 | Page 97

Find the inverse of each of the matrices, if it exists.

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

Q 13 | Page 97

Find the inverse of each of the matrices, if it exists.

[(2,1),(4,2)]

Q 14 | Page 97

Find the inverse of each of the matrices, if it exists.

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

Q 15 | Page 97

Find the inverse of each of the matrices, if it exists.

[(2,0,-1),(5,1,0),(0,1,3)]

Q 15 | Page 97

Find the inverse of each of the matrices, if it exists.

[(1,3,-2),(-3,0,-5),(2,5,0)]

Q 16 | Page 97

Find the inverse of each of the matrices, if it exists.

[(1,3,-2),(-3,0,-5),(2,5,0)]

Q 16 | Page 97

Find the inverse of each of the matrices, if it exists.

[(2,0,-1),(5,1,0),(0,1,3)]

Q 17 | Page 97

Matrices A and B will be inverse of each other only if

A. AB = BA

C. AB = 0, BA = I

B. AB = BA = 0

D. AB = BA = I

Q 18 | Page 97

#### Pages 100 - 101

Let A = [(0,1),(0,0)]show that (aI+bA)n  = anI + nan-1 bA , where I is the identity matrix of order 2 and n ∈ N

Q 1 | Page 100

If A and B are symmetric matrices, prove that AB − BA is a skew symmetric matrix

Q 4 | Page 100

Show that the matrix B'AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.

Q 5 | Page 100

Find the values of xyz if the matrix A = [(0,2y,z),(x,y,-z),(x , -y,z)] satisfy the equation

A'A = I

Q 6 | Page 100

For what values of x, [1,2,1] [(1,2,0),(2,0,1),(1,0,2)][(0),(2),(x)] = O?

Q 7 | Page 100

if A = [(3,1),(-1,2)]  show that  A^2 - 5A + 7I = O

Q 8 | Page 100

Find x, if [x, -5, -1][(1,0,2),(0,2,1),(2,0,3)][(x),(4),(1)] = O

Q 9 | Page 100

A manufacturer produces three products xyz which he sells in two markets.

Annual sales are indicated below:

 Market Products I 10000 2000 18000 II 6000 20000 8000

(a) If unit sale prices of xy and are Rs 2.50, Rs 1.50 and Rs 1.00, respectively, find the total revenue in each market with the help of matrix algebra.

(b) If the unit costs of the above three commodities are Rs 2.00, Rs 1.00 and 50 paise respectively. Find the gross profit.

Q 10 | Page 101

Find the matrix X so that  X[(1,2,3),(4,5,6)]= [(-7,-8,-9),(2,4,6)]

Q 11 | Page 101

If A and B are square matrices of the same order such that AB = BA, then prove by induction that AB" = B"A. Further, prove that (AB)" = A"B" for all n ∈ N

Q 12 | Page 101

Choose the correct answer in the following questions:

if A = [(alpha, beta),(gamma, -alpha)]` is such that A2 = I then

(A) 1 + α² + βγ = 0

(B) 1 – α² + βγ = 0

(C) 1 – α² – βγ = 0

(D) 1 + α² – βγ = 0

Q 13 | Page 101

If the matrix A is both symmetric and skew symmetric, then

A. A is a diagonal matrix

B. A is a zero matrix

C. A is a square matrix

D. None of these

Q 14 | Page 101

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

(A) A

(B) I – A

(C) I

(D) 3A

Q 15 | Page 101

S