If A = [0-1243-4] and B = [401326], then verify that: (kA)' = (kA') - Mathematics

Advertisements
Advertisements
Sum

If A = `[(0, -1, 2),(4, 3, -4)]` and B = `[(4, 0),(1, 3),(2, 6)]`, then verify that: (kA)' = (kA')

Advertisements

Solution

Given that: A = `[(0, -1, 2),(4, 3, -4)]`, B = `[(4, 0),(1, 3),(2, 6)]`

L.H.S. kA = `"k"[(0, -1, 2),(4, 3, -4)]`

= `[(0, -"k", 2"k"),(4"k", 3"k", -4"k")]`

(kA)' = `[(0, 4"k"),(-"k", 3"k"),(2"k", -"k")]`

R.H.S. kA' = `"k"[(0, -1, 2),(4, 3, -4)]^'`

= `"k"[(0, 4),(-1, 3),(2, -4)]`

= `[(0, 4"k"),(-"k", 3"k"),(2"k", -4"k")]`

Hence, L.H.S. = R.H.S.

(kA)' = (kA') is verified.

  Is there an error in this question or solution?
Chapter 3: Matrices - Exercise [Page 56]

APPEARS IN

NCERT Exemplar Mathematics Class 12
Chapter 3 Matrices
Exercise | Q 27. (iii) | Page 56

RELATED QUESTIONS

The cost of 4 pencils, 3 pens and 2 erasers is Rs. 60. The cost of 2 pencils, 4 pens and 6 erasers is Rs. 90 whereas the cost of 6 pencils, 2 pens and 3 erasers is Rs. 70. Find the cost of each item by using matrices.


The sum of three numbers is 6. When second number is subtracted from thrice the sum of first and third number, we get number 10. Four times the sum of third number is subtracted from five times the sum of first and second number, the result is 3. Using above information, find these three numbers by matrix method.


Find the inverse of the matrix,  `A=[[1,3,3],[1,4,3],[1,3,4]]`by using column transformations.


Solve the following equations by the method of reduction :

2x-y + z=1,  x + 2y +3z = 8, 3x + y-4z=1.


The sum of three numbers is 9. If we multiply third number by 3 and add to the second number, we get 16. By adding the first and the third number and then subtracting twice the second number from this sum, we get 6. Use this information and find the system of linear equations. Hence, find the three numbers using matrices.


Express the following equations in the matrix form and solve them by method of reduction :

2x- y + z = 1, x + 2y + 3z = 8, 3x + y - 4z =1


Use elementary column operation C2 → C2 + 2C1 in the following matrix equation :

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


For what values of k, the system of linear equations

x + y + z = 2
2x + y – z = 3
3x + 2y + kz = 4

has a unique solution?

 


If `A=|[2,0,-1],[5,1,0],[0,1,3]|` , then find A-1 using elementary row operations


Using the properties of determinants, solve the following for x:

`|[x+2,x+6,x-1],[x+6,x-1,x+2],[x-1,x+2,x+6]|=0`


Prove that  `|(yz-x^2,zx-y^2,xy-z^2),(zx-y^2,xy-z^2,yz-x^2),(xy-z^2,yz-x^2,zx-y^2)|`is divisible by (x + y + z) and hence find the quotient.


Using properties of determinants, prove that :

`|[1+a,1,1],[1,1+b,1],[1,1,1+c]|=abc + bc + ca + ab`


The cost of 2 books, 6 notebooks and 3 pens is  Rs 40. The cost of 3 books, 4 notebooks and 2 pens is Rs 35, while the cost of 5 books, 7 notebooks and 4 pens is Rs 61. Using this information and matrix method, find the cost of 1 book, 1 notebook and 1 pen separately.


Using elementary row transformations, find the inverse of the matrix A = `[(1,2,3),(2,5,7),(-2,-4,-5)]`


Prove that :

\[\begin{vmatrix}a & a + b & a + 2b \\ a + 2b & a & a + b \\ a + b & a + 2b & a\end{vmatrix} = 9 \left( a + b \right) b^2\]

 


x + y + z + w = 2
x − 2y + 2z + 2w = − 6
2x + y − 2z + 2w = − 5
3x − y + 3z − 3w = − 3


2x − 3z + w = 1
x − y + 2w = 1
− 3y + z + w = 1
x + y + z = 1


2x − y = 5
4x − 2y = 7


In the following matrix equation use elementary operation R2 → R2 + Rand the equation thus obtained:

\[\begin{bmatrix}2 & 3 \\ 1 & 4\end{bmatrix} \begin{bmatrix}1 & 0 \\ 2 & - 1\end{bmatrix} = \begin{bmatrix}8 & - 3 \\ 9 & - 4\end{bmatrix}\]

Use elementary column operation C2 → C2 + 2C1 in the following matrix equation : \[\begin{bmatrix} 2 & 1 \\ 2 & 0\end{bmatrix} = \begin{bmatrix}3 & 1 \\ 2 & 0\end{bmatrix}\begin{bmatrix}1 & 0 \\ - 1 & 1\end{bmatrix}\]


Use elementary column operations  \[C_2 \to C_2 - 2 C_1\] in the matrix equation \[\begin{pmatrix}4 & 2 \\ 3 & 3\end{pmatrix} = \begin{pmatrix}1 & 2 \\ 0 & 3\end{pmatrix}\begin{pmatrix}2 & 0 \\ 1 & 1\end{pmatrix}\] .


If three numbers are added, their sum is 2. If two times the second number is subtracted from the sum of the first and third numbers, we get 8, and if three times the first number is added to the sum of the second and third numbers, we get 4. Find the numbers using matrices. 


Using elementary row operations, find the inverse of the matrix A = `((3, 3,4),(2,-3,4),(0,-1,1))` and hence solve the following system of equations :  3x - 3y + 4z = 21, 2x -3y + 4z = 20, -y + z = 5.


Apply the given elementary transformation on each of the following matrices `[(3, -4),(2, 2)]`, R1 ↔ R2.


Transform `[(1, -1, 2),(2, 1, 3),(3, 2, 4)]` into an upper traingular matrix by suitable row transformations.


Find the cofactor matrix, of the following matrices : `[(1, 2),(5, -8)]`


Find the cofactor matrix, of the following matrices: `[(5, 8, 7),(-1, -2, 1),(-2, 1, 1)]`


Find the adjoint of the following matrices : `[(2, -3),(3, 5)]`


Find the adjoint of the following matrices : `[(1, -1, 2),(-2, 3, 5),(-2, 0, -1)]`


Choose the correct alternative.

If A = `[("a", 0, 0),(0, "a", 0),(0, 0,"a")]`, then |adj.A| = _______


Choose the correct alternative.

If A = `[(2, 5),(1, 3)]`, then A–1 = _______


Fill in the blank :

Order of matrix `[(2, 1, 1),(5, 1, 8)]` is _______


If three numbers are added, their sum is 2. If 2 times the second number is subtracted from the sum of first and third numbers, we get 8. If three times the first number is added to the sum of second and third numbers, we get 4. Find the numbers using matrices.


Choose the correct alternative:

If A = `[(1, 2),(2, -1)]`, then adj (A) = ______


Matrix `[("a", "b", "c"),("p", "q", "r"),(2"a" - "p", 2"b" - "q", 2"c" - "r")]` is a singular


State whether the following statement is True or False:

After applying elementary transformation R1 – 3R2 on matrix `[(3, -2),(1, 4)]` we get `[(0, -12),(1, 4)]`


The suitable elementary row transformation which will reduce the matrix `[(1, 0),(2, 1)]` into identity matrix is ______


Find the inverse of matrix A = `[(1, 0, 1),(0, 2, 3),(1, 2, 1)]` by using elementary row transformations 


For which values of xis the matrix

`[(3,-1+x,2),(3,-1,x+2),(x+3,-1,2)]` non-invertible?


If `overlinea = 3hati + hatj + 4hatk, overlineb = 2hati - 3hatj + lambdahatk, overlinec = hati + 2hatj - 4hatk` and `overlinea.(overlineb xx overlinec) = 47`, then λ is equal to ______


If A = `[(a, 0, 0), (0, a, 0), (0, 0, a)]`, then the value of |A| |adj A| is ______ 


If AX = B, where A = `[(1, 2, 3), (-1, 1, 2), (1, 2, 4)]` and B = `[(1), (2), (3)]`, then X is equal to ______


If A = `[(1, 1, -1), (1, -2, 1), (2, -1, -3)]`, then (adj A)A = ______


Let F(α) = `[(cosalpha, -sinalpha, 0), (sinalpha, cosalpha, 0), (0, 0, 1)]` where α ∈ R. Then [F(α)]-1 is equal to ______ 


If `[(2, 3), (3, 1)][(x), (y)] = [(-5), (3)]`, then the values of x and y respectively are ______


The inverse of a symmetric matrix is ______.


If a matrix has 28 elements, what are the possible orders it can have? What if it has 13 elements?


In the matrix A = `[("a", 1, x),(2, sqrt(3), x^2 - y),(0, 5, (-2)/5)]`, write: The number of elements


Construct a 3 × 2 matrix whose elements are given by aij = ei.x sinjx.


Find the values of a and b if A = B, where A = `[("a" + 4, 3"b"),(8, -6)]`, B = `[(2"a" + 2, "b"^2 + 2),(8, "b"^2 - 5"b")]`


Find A, if `[(4),(1),(3)]` A = `[(-4, 8,4),(-1, 2, 1),(-3, 6, 3)]`


If possible, find BA and AB, where A = `[(2, 1, 2),(1, 2, 4)]`, B = `[(4, 1),(2, 3),(1, 2)]`


Solve for x and y: `x[(2),(1)] + y[(3),(5)] + [(-8),(-11)]` = O


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 = `[(x"a", 0, 0),(0, y"b", 0),(0, 0, z"c")]` = QP


If A = `[(0, -1, 2),(4, 3, -4)]` and B = `[(4, 0),(1, 3),(2, 6)]`, then verify that: (A′)′ = A


If A = `[(0, -1, 2),(4, 3, -4)]` and B = `[(4, 0),(1, 3),(2, 6)]`, then verify that: (A′)′ = (AB)' = B'A'


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


Find the values of a, b, c and d, if `3[("a", "b"),("c", "d")] = [("a", 6),(-1, 2"d")] + [(4, "a" + "b"),("c" + "d", 3)]`


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


Find x, y, z if A = `[(0, 2y, z),(x, y, -z),(x, -y, z)]` satisfies A′ = A–1.


If possible, using elementary row transformations, find the inverse of the following matrices

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


If possible, using elementary row transformations, find the inverse of the following matrices

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


If `[(2x + y, 4x),(5x - 7, 4x)] = [(7, 7y - 13),(y, x + 6)]`, then the value of x + y is ______.


If A = `1/pi [(sin^-1(xpi), tan^-1(x/pi)),(sin^-1(x/pi), cot^-1(pix))]`, B = `1/pi [(-cos^-1(x/pi), tan^-1 (x/pi)),(sin^-1(x/pi),-tan^-1(pix))]`, then A – B is equal to ______.


On using elementary column operations C2 → C2 – 2C1 in the following matrix equation `[(1, -3),(2, 4)] = [(1, -1),(0, 1)] [(3, 1),(2, 4)]`, we have: ______.


On using elementary row operation R1 → R1 – 3R2 in the following matrix equation: `[(4, 2),(3, 3)] = [(1, 2),(0, 3)] [(2, 0),(1, 1)]`, we have: ______.


In applying one or more row operations while finding A–1 by elementary row operations, we obtain all zeros in one or more, then A–1 ______.


A matrix denotes a number.


Two matrices are equal if they have same number of rows and same number of columns.


If A = `[(2, 3, -1),(1, 4, 2)]` and B = `[(2, 3),(4, 5),(2, 1)]`, then AB and BA are defined and equal.


If A = `[(0,0,0,0),(0,0,0,0),(1,0,0,0),(0,1,0,0)],` then ____________.


If `[(2, 0, 7),(0, 1, 0),(1, -2, 1)] [(-x, 14x, 7x),(0, 1, 0),(x, -4x, -2x)] = [(1, 0, 0),(0, 1, 0),(0, 0, 1)]`then find the value of x


If f(x) = `|(1 + sin^2x, cos^2x, 4 sin 2x),(sin^2x, 1 + cos^2x, 4 sin 2x),(sin^2 x, cos^2 x, 1 + 4 sin 2x)|` 

What is the maximum value of f(x)?


If `[(3,0),(0,2)][(x),(y)] = [(3),(2)], "then"  x = 1  "and"  y = -1`


Share
Notifications



      Forgot password?
Use app×