Advertisements
Advertisements
Question
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'
Advertisements
Solution
Given, A = `[(-1, 2, 3),(5, 7, 9),(-2, 1, 1)]` and B = `[(-4, 1, -5),(1, 2, 0),(1, 3, 1)]`
Then, (A – B) = A = `[(-1, 2, 3),(5, 7, 9),(-2, 1, 1)] - [(-4, 1, -5),(1, 2, 0),(1, 3, 1)]`
= `[(-1 + 4, 2 - 1, 3 + 5),(5 - 1, 7 - 2, 9 - 0), (-2 - 1, 1 - 3, 1 - 1)]`
= `[(3, 1, 8),(4, 5, 9),(-3, -2, 0)]`
Then, (A – B)' = `[(3, 1, 8),(4, 5, 9),(-3, -2, 0)] = [(3, 4, -3),(1, 5, -2),(8, 9, 0)] ` ...(i)
We know that, A' = `[(-1, 5, -2), (2, 7, 1),(3, 9, 1)]` and B' = `[(-4, 1, 1),(1, 2, 3),(-5, 0, 1)]`
A' – B' = `[(-1, 5, -2), (2, 7, 1),(3, 9, 1)] - [(-4, 1, 1),(1, 2, 3),(-5, 0, 1)]`
= `[(-1 + 4, 5 - 1, -2 - 1),(2 - 1, 7 - 2, 1 - 3),(3 + 5, 9 - 0, 1 - 1)]`
= `[(3, 4, -3),(1, 5, -2),(8, 9, 0)]` ...(ii)
Equations (i) and (ii) prove that,
(A – B)' = A' – B'
APPEARS IN
RELATED QUESTIONS
Matrix A = `[(0,2b,-2),(3,1,3),(3a,3,-1)]`is given to be symmetric, find values of a and b
If A' = `[(3, 4),(-1, 2),(0, 1)]` and B = `[(-1, 2, 1),(1, 2, 3)]`, then verify that (A – B)' = A' – B'
If A' = `[(-2, 3),(1, 2)]` and B = `[(-1, 0),(1, 2)]`, then find (A + 2B)'
For the matrices A and B, verify that (AB)′ = B'A', where A = `[(1),(-4),(3)]`, B = `[(-1, 2, 1)]`
Show that the matrix A = `[(1, -1, 5),(-1, 2, 1),(5, 1, 3)]` is a symmetric matrix.
Show that the matrix A = `[(0, 1, -1),(-1, 0, 1),(1, -1, 0)]` is a skew symmetric matrix.
Find `1/2` (A + A') and `1/2` (A – A'), when A = `[(0, a, b),(-a, 0, c),(-b, -c, 0)]`
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:
`[(6, -2, 2),(-2, 3, -1),(2, -1, 3)]`
Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
`[(1, 5),(-1, 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.
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.
For what value of x, is the matrix \[A = \begin{bmatrix}0 & 1 & - 2 \\ - 1 & 0 & 3 \\ x & - 3 & 0\end{bmatrix}\] a skew-symmetric matrix?
If A = [aij] is a square matrix of even order such that aij = i2 − j2, then
If A and B are matrices of the same order, then ABT − BAT is a
The matrix \[A = \begin{bmatrix}1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 4\end{bmatrix}\] is
If the matrix `((6,-"x"^2),(2"x"-15 , 10))` is symmetric, find the value of x.
Express the matrix `[(2, 3, 1),(1, -1, 2),(4, 1, 2)]` as the sum of a symmetric and a skew-symmetric matrix.
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 ______.
______ matrix is both symmetric and skew-symmetric matrix.
Sum of two skew-symmetric matrices is always ______ matrix.
If A is a skew-symmetric matrix, then A2 is a ______.
If A is skew-symmetric, then kA is a ______. (k is any scalar)
If A and B are symmetric matrices, then BA – 2AB is a ______.
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 ____________.
If A = `[(3, "x" - 1),(2"x" + 3, "x" + 2)]` is a symmetric matrix, then x = ____________.
If A `= [(6,8,5),(4,2,3),(9,7,1)]` is the sum of a symmetric matrix B and skew-symmetric matrix C, then B is ____________.
If A, B are Symmetric matrices of same order, then AB – BA is a
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 ______.
The value of |A|, if A = `[(0, 2x - 1, sqrt(x)),(1 - 2x, 0, 2sqrt(x)),(-sqrt(x), -2sqrt(x), 0)]`, where x ∈ R+, is ______.
