Question

Question: If A and B are arbitrary n × n matrices, which of the matrices in Exercise 21 through 26 must be symmetric?

${\mathit{A}}^{\mathbf{T}}{\mathit{B}}^{\mathbf{T}}\mathit{B}\mathit{A}$.

Short answer

$A^T{B}^{T}BA$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,

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

Thus,

$LetC=A^T{B}^T{\left(A^T\right)}^T{\left(B^T\right)}^T$

Then,

$C=A^T{B}^T{\left(A^T\right)}^T{\left(B^T\right)}^T\Rightarrow C^T=A^T{B}^{T}BAC=C^T$

Therefore,If A and B are arbitraryn × n matrices then the matrices $A^T{B}^{T}BA$is symmetric.

Hence, the matrices is symmetric.