If $A$ is a $3 \times 3$ order skew-symmetric matrix,then $|A|$ is equal to:

For any square matrix $A$,$AA^T$ is a

If $A=\left[\begin{array}{rrr}-1 & 2 & 3 \\ 5 & 7 & 9 \\ -2 & 1 & 1\end{array}\right]$ and $B=\left[\begin{array}{rrr}-4 & 1 & -5 \\ 1 & 2 & 0 \\ 1 & 3 & 1\end{array}\right],$ then verify that $(A-B)^{\prime}=A^{\prime}-B^{\prime}$.

If $A$ and $B$ are symmetric matrices,prove that $AB - BA$ is a skew-symmetric matrix.

Show that the matrix $B^{\prime}AB$ is symmetric or skew-symmetric according as $A$ is symmetric or skew-symmetric.

In a skew-symmetric matrix,the diagonal elements are all