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'
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 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.
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.
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`
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
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, 1, -1),(1, 3, 1),(-1, 1, 3)]`, 3R2 and C2 ↔ C2 – 4C1.
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 : `[(1, -1, 2),(-2, 3, 5),(-2, 0, -1)]`
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 _______
State whether the following is True or False :
Single element matrix is row as well as column matrix.
Solve the following :
If A = `[(1, 0, 0),(2, 1, 0),(3, 3, 1)]`, the reduce it to unit matrix by using row transformations.
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)]`
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.
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 `[(2, 3), (3, 1)][(x), (y)] = [(-5), (3)]`, then the values of x and y respectively are ______
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 ______.
In the matrix A = `[("a", 1, x),(2, sqrt(3), x^2 - y),(0, 5, (-2)/5)]`, write: elements a23, a31, a12
Find the matrix A satisfying the matrix equation:
`[(2, 1),(3, 2)] "A" [(-3, 2),(5, -3)] = [(1, 0),(0, 1)]`
Find A, if `[(4),(1),(3)]` A = `[(-4, 8,4),(-1, 2, 1),(-3, 6, 3)]`
Solve for x and y: `x[(2),(1)] + y[(3),(5)] + [(-8),(-11)]` = O
If `[(xy, 4),(z + 6, x + y)] = [(8, w),(0, 6)]`, then find values of x, y, z and w.
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.
If P(x) = `[(cosx, sinx),(-sinx, cosx)]`, then show that P(x) . (y) = P(x + y) = P(y) . P(x)
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, 3, -3),(-1, 2, 2),(1, 1, -1)]`
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 (AB)′ = B′ A′, where A and B are not square matrices, then number of rows in A is equal to number of columns in B and number of columns in A is equal to number of rows in B.
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 `[(3,0),(0,2)][(x),(y)] = [(3),(2)], "then" x = 1 "and" y = -1`
