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Found in: Page 233

### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

# If A and B are arbitrary ${\mathbit{n}}{\mathbf{×}}{\mathbit{n}}$ matrices, which of the matrices in Exercise 21 through 26 must be symmetric?${{\mathbit{A}}}^{{\mathbf{\Gamma }}}{\mathbit{B}}{\mathbit{A}}$.

The matrix ${A}^{T}BA$ is not symmetric.

See the step by step solution

## Step 1: Condition to be a symmetric.

A matrix is symmetric if and only if it is equal to its transpose.

All entries above the main diagonal of a symmetric matrix are reflected into equal entries below the diagonal. A matrix is skew-symmetric if and only if it is the opposite of its transpose.

For example,

$\left(\begin{array}{ccc}1& 1& -1\\ 1& 2& 0\\ -1& 0& 5\end{array}\right)$

Thus, it gives

${\left({A}^{T}BA\right)}^{T}={\left(BA\right)}^{T}{\left({A}^{T}\right)}^{T}\phantom{\rule{0ex}{0ex}}={A}^{T}{B}^{T}A$

Therefore, if A and B are arbitrary $n×n$ matrices then the matrices ${A}^{T}BA$ is not symmetric.

Hence, the matrix is not necessarily symmetric.