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Q15E

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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 the nxn matrices A and B are symmetric and B is invertible, which of the matrices in Exercise 13 through 20 must be symmetric as well?AB.

The Matrix AB is not symmetric.

See the step by step solution

## Step 1: Definition of symmetric matrix.

A square matrix is symmetric matrix if ${\mathbf{A}}{\mathbf{=}}{{\mathbf{A}}}^{{\mathbf{T}}}$

## Step 2: Verification whether the given matrix is symmetric.

Given that A and B are symmetric matrices and B is invertible.

Since A is symmetric, then $A={A}^{T}$.

Then by properties of transpose, ${\left(\mathrm{AB}\right)}^{\mathrm{T}}={\mathrm{B}}^{\mathrm{T}}{\mathrm{A}}^{\mathrm{T}}$it gives,

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

Therefore,AB is not a symmetric matrix.