Advertisements
Advertisements
Question
If A = `[(0, -1, 2),(4, 3, -4)]` and B = `[(4, 0),(1, 3),(2, 6)]`, then verify that: (A′)′ = (AB)' = B'A'
Advertisements
Solution
Given that: A = `[(0, -1, 2),(4, 3, -4)]`, B = `[(4, 0),(1, 3),(2, 6)]`
L.H.S. AB = `[(0,-1, 2),(4, 3, -4)]_(2 xx 3) [(4, 0),(1, 3),(2, 6)]_(3 xx 2)`
= `[(0 - 1+ 4, 0 - 3 + 12),(16 + 3 - 8, 0 + 9 - 24)]_(2 xx 2)`
= `[(3, 9),(11, -15)]_(2 xx 2)`
(AB)' = `[(3, 11),(9, -15)]_(2 xx 2)`
R.H.S. B' = `[(4, 0),(1, 3),(2, 6)]^'`
= `[(4, 1, 2),(0, 3, 6)]`
A' = `[(0, -1, 2),(4, 3, -4)]^'`
= `[(0, 4),(-1, 3),(2, -4)]`
B'A' = `[(4, 1, 2),(0, 3, 6)]_(2 xx 3) [(0, 4),(-1, 3),(2, -4)]_(3 xx 2)`
= `[(0 - 1 + 4, 16 + 3 - 8),(0 - 3 + 12, 0 + 9 - 24)]_(2 xx 2)`
= `[(3, 11),(9, -15)]_(2 xx 2)`
L.H.S. = R.H.S.
Hence, (AB)' = B'A' is verified.
APPEARS IN
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.
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
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 elementary transformations, find the inverse of the matrix A = `((8,4,3),(2,1,1),(1,2,2))`and use it to solve the following system of linear equations :
8x + 4y + 3z = 19
2x + y + z = 5
x + 2y + 2z = 7
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 :
2x − 3z + w = 1
x − y + 2w = 1
− 3y + z + w = 1
x + y + z = 1
2x − y = 5
4x − 2y = 7
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.
Apply the given elementary transformation on each of the following matrices `[(3, -4),(2, 2)]`, R1 ↔ R2.
Apply the given elementary transformation on each of the following matrices `[(2, 4),(1, -5)]`, C1 ↔ C2.
Apply the given elementary transformation on each of the following matrices `[(3, 1, -1),(1, 3, 1),(-1, 1, 3)]`, 3R2 and C2 ↔ C2 – 4C1.
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| = _______
Fill in the blank :
Order of matrix `[(2, 1, 1),(5, 1, 8)]` is _______
Solve the following :
If A = `[(1, 0, 0),(2, 1, 0),(3, 3, 1)]`, the reduce it to unit matrix by using row transformations.
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.
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 A is a 3 × 3 matrix and |A| = 2, then the matrix represented by A (adj A) is equal to.
The cofactors of the elements of the first column of the matrix A = `[(2,0,-1),(3,1,2),(-1,1,2)]` are ______.
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 `overlinea = hati + hatj + hatk, overlinea . overlineb = 1` and `overlinea xx overlineb = hatj - hatk,` then `overlineb` = ______
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 A = `[(1, 2, 1), (3, 2, 3), (2, 1, 2)]`, then `a_11A_11 + a_21A_21 + a_31A_31` = ______
The inverse of a symmetric matrix is ______.
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 non-zero values of x satisfying the matrix equation:
`x[(2x, 2),(3, x)] + 2[(8, 5x),(4, 4x)] = 2[(x^2 + 8, 24),(10, 6x)]`
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
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 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: ______.
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 ____________.
`abs((1,1,1),("e",0,sqrt2),(2,2,2))` is equal to ____________.
if `A = [(2,5),(1,3)] "then" A^-1` = ______
