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Q14E

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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? -B.

The Matrix -B is symmetric.

See the step by step solution

## Step 1: Definition of symmetric matrix.

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

## 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 $\mathrm{A}={\mathrm{A}}^{\mathrm{T}}$.

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

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

Therefore, -B is a symmetric matrix.

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