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?${\mathbf{AB}}^{\mathbf{2}}\mathbf{A}$.

Short answer

The Matrix${\mathrm{AB}}^2\mathrm{A}$ is symmetric.

Step-by-step solution

Definition of symmetric matrix.

A square matrix is symmetric matrix if$\mathit{A}{\mathit{B}}^{\mathbf{2}}\mathbf{A}$

Verification whether the given matrix is symmetric.

Given that A and B are symmetric matrices and B is invertible.

Since A is symmetric, then $A=A^T$.

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

$\begin{array}{r}{\left({\mathrm{AB}}^2\mathrm{A}\right)}^T=(ABBA{)}^T\\ =\left[\right(AB\left)\right(BA){]}^T\\ =\left(BA{)}^T\right(AB{)}^T\\ =\left(A^T{B}^T\right)\left(B^T{A}^T\right)\end{array}$

Further solve the above expression,

$$\begin{gathered} {\left({\mathrm{AB}}^2\mathrm{A}\right)}^T=\left(AB\right)\left(BA\right) \\ =A\left(BB\right)A \\ ={\mathrm{AB}}^2\mathrm{A} \end{gathered}$$

Therefore,${\mathrm{AB}}^2\mathrm{A}$ is a symmetric matrix.