If $A$ is a square matrix and $A + A^T$ is a symmetric matrix,then $A - A^T$ is:

If $A$ and $B$ are square matrices of the same order,then which of the following properties holds true for the transpose of their product?

$A=\begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}$ and $B=\begin{bmatrix} x & y \\ 1 & 2 \end{bmatrix}$ are two matrices such that $(A+B)(A-B)=A^2-B^2$. If $C=\begin{bmatrix} x & 2 \\ 1 & y \end{bmatrix}$,then $\operatorname{Trace}(C)=$

If $A^{\prime}=\begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 & 1 \end{bmatrix}$ and $B=\begin{bmatrix} -1 & 2 & 1 \\ 1 & 2 & 3 \end{bmatrix}$,then verify that $(A+B)^{\prime}=A^{\prime}+B^{\prime}$.

If $A$ and $B$ are symmetric matrices of the same order,then show that $AB$ is symmetric if and only if $A$ and $B$ commute,that is $AB = BA$.

Express the matrix $A = \left[\begin{array}{cc}1 & 5 \\ -1 & 2\end{array}\right]$ as the sum of a symmetric and a skew-symmetric matrix.