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?${\mathbf{A}}^{\mathbf{10}}$.

Short answer

The Matrix${\mathrm{A}}^{10}$ 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,$\left(A{B}^T\right)=B^T{A}^T$ it gives,

$$\begin{gathered} {\left(A^{10}\right)}^T={\left(AAAAAAAAAA\right)}^T \\ =A^T{A}^T{A}^T{A}^T{A}^T{A}^T{A}^T{A}^T{A}^T{A}^T \\ ={\left(A^T\right)}^{10} \\ ={\left(A\right)}^{10} \end{gathered}$$

Therefore,$A^{10}$ is a symmetric matrix.