Advertisements
Advertisements
प्रश्न
For the matrices A and B, verify that (AB)′ = B'A' where `A =[(1),(-4), (3)], B = [-1, 2 1]`
उत्तर
Given, `"A" = [(1),(-4), (3)], "B" = [(-1, 2, 1)]`
So, AB = `[(1),(-4), (3)] xx [(-1, 2, 1)]`
`= [(1 xx (-1), 1 xx 2, 1 xx 1), (-4 xx (-1), -4 xx 2, -4 xx 1),(3 xx (-1), 3 xx 2, 3 xx 1)]`
`= [(-1, 2, 1), (4, -8, -4), (-3,6,3)]`
Now, (AB)' = `[(-1, 4, -3),(2,-8,6), (1, -4, 3)]` ....(i)
A' `= [(1, -4, 3)]` and B' `= [(-1),(2),(1)]`
Now, B'A' = `[(-1),(2),(1)] xx [(1, -4, 3)]`
`= [(-1 xx 1, -1 xx (-4), -1 xx 3),(2 xx 1, 2 xx (-4), 2 xx 3), (1 xx 1, 1 xx (-4), 1 xx 3)]`
`= [(-1, 4, -3),(2,-8,6),(1,-4,3)]` ....(ii)
It is proved from the equation and that, (AB)' = B'A'
APPEARS IN
संबंधित प्रश्न
Matrix A = `[(0,2b,-2),(3,1,3),(3a,3,-1)]`is given to be symmetric, find values of a and b
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'
if A' = `[(-2,3),(1,2)] and B = [(-1,0),(1,2)]` then find (A + 2B)'
If A = `[(sin alpha, cos alpha), (-cos alpha, sin alpha)]` then verify that A'A = I
Show that the matrix A = `[(0,1,-1),(-1,0,1),(1,-1,0)]` is a skew symmetric matrix.
For the matrix A = `[(1,5),(6,7)]` verify that (A + A') is a symmetric matrix.
Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
`[(3,5),(1,-1)]`
Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
`[(3,3,-1),(-2,-2,1),(-4,-5,2)]`
If A and B are symmetric matrices, prove that AB − BA is a skew symmetric matrix.
Show that the matrix B'AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.
Show that all the diagonal elements of a skew symmetric matrix are zero.
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.
Write a square matrix which is both symmetric as well as skew-symmetric.
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.
If A is a square matrix, then AA is a
If A and B are symmetric matrices, then ABA is
The matrix \[A = \begin{bmatrix}0 & - 5 & 8 \\ 5 & 0 & 12 \\ - 8 & - 12 & 0\end{bmatrix}\] is a
The matrix \[A = \begin{bmatrix}1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 4\end{bmatrix}\] is
If A = `[(0, 1),(1, 1)]` and B = `[(0, -1),(1, 0)]`, show that (A + B)(A – B) ≠ A2 – B2
If the matrix `[(0, "a", 3),(2, "b", -1),("c", 1, 0)]`, is a skew symmetric matrix, find the values of a, b and c.
The matrix `[(1, 0, 0),(0, 2, 0),(0, 0, 4)]` is a ______.
The matrix `[(0, -5, 8),(5, 0, 12),(-8, -12, 0)]` is a ______.
If A is a symmetric matrix, then A3 is a ______ matrix.
If A and B are symmetric matrices, then AB – BA is a ______.
If A and B are symmetric matrices, then BA – 2AB is a ______.
If A is symmetric matrix, then B′AB is ______.
AA′ is always a symmetric matrix for any matrix A.
If A is skew-symmetric matrix, then A2 is a symmetric matrix.
If A and B are symmetric matrices of the same order, then ____________.
The diagonal elements of a skew symmetric matrix are ____________.
Let A and B be and two 3 × 3 matrices. If A is symmetric and B is skewsymmetric, then the matrix AB – BA is ______.
If `[(2, 0),(5, 4)]` = P + Q, where P is symmetric, and Q is a skew-symmetric matrix, then Q is equal to ______.
Number of symmetric matrices of order 3 × 3 with each entry 1 or – 1 is ______.
If A and B are symmetric matrices of the same order, then AB – BA is ______.
