• :00Days
• :00Hours
• :00Mins
• 00Seconds
A new era for learning is coming soon Suggested languages for you:

Europe

Answers without the blur. Sign up and see all textbooks for free! Q16E

Expert-verified 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?A+B.

The Matrix A + B is symmetric.

See the step by step solution

## Step 1: Definition of symmetric matrix.

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

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

Therefore, A + B is a symmetric matrix. ### Want to see more solutions like these? 