Advertisements
Advertisements
Question
If A and B are symmetric matrices, then ABA is
Options
symmetric matrix
skew-symmetric matrix
diagonal matrix
scalar matrix
Advertisements
Solution
symmetric matrix
since A and B are symmetric matrices, we get
` A =A ^' and B =B^' `
\[\left( ABA \right)' = \left( BA \right)' \left( A \right)' \]
\[ = A'B'A'\]
\[ = ABA \left[ \because A =\text{ A' and B} = B' \right]\]
\[Since \left ( ABA \right)' = ABA, ABA \text{ is a symmetric matrix} .\]
RELATED QUESTIONS
If A= `((3,5),(7,9))`is written as A = P + Q, where P is a symmetric matrix and Q is skew symmetric matrix, then write the matrix P.
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 = `[(cos α, sin α), (-sin α, cos α)]`, then verify that A' A = I
If A = `[(sin α, cos α), (-cos α, sin α)]`, 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.
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)]`
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.
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 is a square matrix, then AA is a
If A = [aij] is a square matrix of even order such that aij = i2 − j2, then
If A and B are two matrices of order 3 × m and 3 × n respectively and m = n, then the order of 5A − 2B is
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
Let A = `[(2, 3),(-1, 2)]`. Then show that A2 – 4A + 7I = O. Using this result calculate A5 also.
If A = `[(0, 1),(1, 1)]` and B = `[(0, -1),(1, 0)]`, show that (A + B)(A – B) ≠ A2 – B2
Show that A′A and AA′ are both symmetric matrices for any matrix A.
If A = `[(cosalpha, sinalpha),(-sinalpha, cosalpha)]`, and A–1 = A′, find value of α
Express the matrix `[(2, 3, 1),(1, -1, 2),(4, 1, 2)]` as the sum of a symmetric and a skew-symmetric matrix.
______ matrix is both symmetric and skew-symmetric matrix.
If A is a skew-symmetric matrix, then A2 is a ______.
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 and B are symmetric matrices of same order, then AB is symmetric if and only if ______.
If each of the three matrices of the same order are symmetric, then their sum is a symmetric matrix.
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 and B are symmetric matrices of the same order, then ____________.
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 ______.
