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?AB.

Short answer

The Matrix AB is not 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 $A=A^T$.

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

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

Therefore,AB is not a symmetric matrix.