Question

If the nxn matrices Aand Bare symmetric and Bis invertible, which of the matrices in Exercise 13 through 20 must be symmetric as well? -B.

Short answer

The Matrix -B is symmetric.

Step-by-step solution

Definition of symmetric matrix.

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

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,$\left(\mathrm{kA}\right)={\mathrm{kA}}^{\mathrm{T}}$ it gives,

$$\begin{gathered} {\left(-\mathrm{B}\right)}^{\mathrm{T}}=-{\mathrm{B}}^{\mathrm{T}} \\ =-\mathrm{B} \end{gathered}$$

Therefore, -B is a symmetric matrix.