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?A+B.

Short answer

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

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

Therefore, A + B is a symmetric matrix.