Advertisements
Advertisements
Question
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.
Advertisements
Solution
\[\left( AB - BA \right)^T = \left( AB \right)^T - \left( BA \right)^T \]
\[ \Rightarrow \left( AB - BA \right)^T = B^T A^T - A^T B^T \left[ \because \left( AB \right)^T = B^T A^T \right]\]
\[ \Rightarrow \left( AB - BA \right)^T = BA - AB \left[ \because B^T = \text{B and} A^T = A \right]\]
\[ \Rightarrow \left( AB - BA \right)^T = - \left( AB - BA \right)\]
Therefore, AB - BA is skew - symmetric .
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 is a skew symmetric matric of order 3, then prove that det A = 0
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 = `[(-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' = `[(3, 4),(-1, 2),(0, 1)]` and B = `[(-1, 2, 1),(1, 2, 3)]`, then verify that (A + B)' = A' + B'
For the matrices A and B, verify that (AB)′ = B'A', where A = `[(1),(-4),(3)]`, B = `[(-1, 2, 1)]`
For the matrices A and B, verify that (AB)′ = B'A' where A = `[(0),(1),(2)]`, B = `[(1, 5, 7)]`
For the matrix A = `[(1, 5),(6, 7)]`, verify that (A – A') is a skew symmetric matrix.
Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
`[(6, -2, 2),(-2, 3, -1),(2, -1, 3)]`
Show that the matrix B'AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.
If the matrix A is both symmetric and skew symmetric, then ______.
Write a square matrix which is both symmetric as well as skew-symmetric.
If A = [aij] is a square matrix of even order such that aij = i2 − j2, then
If A and B are symmetric matrices of the same order, then (AB′ –BA′) is a ______.
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 ______.
______ 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 is symmetric matrix, then B′AB is ______.
If A and B are symmetric matrices of same order, then AB is symmetric if and only if ______.
If A and B are any two matrices of the same order, then (AB)′ = A′B′.
AA′ is always a symmetric matrix for any matrix A.
If A is skew-symmetric matrix, then A2 is a symmetric matrix.
If A = `[(3, "x" - 1),(2"x" + 3, "x" + 2)]` is a symmetric matrix, then x = ____________.
If A, B are Symmetric matrices of same order, then AB – BA is a
If A = [aij] is a skew-symmetric matrix of order n, then ______.
If `[(2, 0),(5, 4)]` = P + Q, where P is symmetric, and Q is a skew-symmetric matrix, then Q is equal to ______.
A square matrix \[A=[a_{ij}]_{n\times n}\] is symmetric when which condition holds?
A square matrix \[A=[a_{ij}]_{n\times n}\] is skew-symmetric when which condition holds?
If \[C=A-A^T\] for a square matrix \[A\], what is \[C^T\]?
Which transpose property justifies the step \[(A+A^T)^T=A^T+(A^T)^T\]?
Which expression writes any square matrix \[A\] as the sum of a symmetric and a skew-symmetric matrix?
For \[B=\begin{bmatrix}2&-2&-4\\-1&3&4\\1&-2&-3\end{bmatrix}\], what is its symmetric part \[P=\frac{1}{2}(B+B^T)\]?
For the matrices \[P=\frac{1}{2}(B+B^T)\] and \[Q=\frac{1}{2}(B-B^T)\], what is \[P+Q\]?
How many decompositions of a square matrix into a symmetric part and a skew-symmetric part are possible?
