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Q17E

Expert-verifiedFound in: Page 233

Book edition
5th

Author(s)
Otto Bretscher

Pages
442 pages

ISBN
9780321796974

**Question: If the ${\mathbf{n}}{\mathbf{\times}}{\mathbf{n}}$ ****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.

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

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.

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