Question

If A and B are arbitrary$\mathit{n}\mathbf{\times }\mathit{n}$ matrices, which of the matrices in Exercise 21 through 26 must be symmetric?

$\mathit{B}\mathbf{(}\mathbf{A}\mathbf{+}{\mathbf{A}}^{\mathbf{\Gamma }}\mathbf{)}{\mathit{B}}^{\mathbf{\Gamma }}$.

Short answer

$B(\mathrm{A}+{\mathrm{A}}^{\mathrm{\Gamma }})B^{\mathrm{\Gamma }}$is symmetric.

Step-by-step solution

Condition to be a symmetric.

A matrix is symmetric if and only if it is equal to its transpose.

All entries above the main diagonal of a symmetric matrix are reflected into equal entries below the diagonal. A matrix is skew-symmetric if and only if it is the opposite of its transpose.

For example,

$\left(\begin{array}{ccc}1& 1& -1\\ 1& 2& 0\\ -1& 0& 5\end{array}\right)$

Let $C^{}=B\left(A^{}+A^T\right)B^T$now the transpose of C is found as.

$$\begin{gathered} C=B{\left(A+A^T\right)}^T{B}^T \\ \Rightarrow C^T=B{\left(A^T+A\right)}^T \\ \Rightarrow C=C^T \end{gathered}$$

Hence, the matrix is symmetric.