Question

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

Short answer

The Matrix is A + B symmetric.

Step-by-step solution

Definition of symmetric matrix.

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

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

$$\begin{gathered} {\left(A+B\right)}^T=A^T+B^T \\ =A+B \end{gathered}$$

Therefore, A + B is a symmetric matrix.