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Q17E

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Found in: Page 233

### Linear Algebra With Applications

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

# Question: If the ${\mathbf{n}}{\mathbf{×}}{\mathbf{n}}$ matrices A and B are symmetric and B is invertible, which of the matrices in Exercise 13 through 20 must be symmetric as well?role="math" localid="1659492178067" ${{\mathbf{B}}}^{\mathbf{-}\mathbf{1}}$.

The Matrix is A + B 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, $\mathrm{A}={\mathrm{A}}^{\mathrm{T}}$.

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

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

Therefore, A + B is a symmetric matrix.