If $A = \begin{bmatrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{bmatrix}$,then $A A^T$ is a

If $P$ is a $3 \times 3$ matrix such that $P^{\top}=2 P+I$,where $P^{\top}$ is the transpose of $P$ and $I$ is the $3 \times 3$ identity matrix,then there exists a column matrix $X=\left[\begin{array}{l}x \\ y \\ z\end{array}\right] \neq\left[\begin{array}{l}0 \\ 0 \\ 0\end{array}\right]$ such that

Let $A$ be a skew-symmetric matrix of odd order,then $|A|$ is equal to

An orthogonal matrix is

Find the transpose of the following matrix: $\left[\begin{array}{c}5 \\ \frac{1}{2} \\ -1\end{array}\right]$.

For the matrix $A = \begin{bmatrix} 1 & 5 \\ 6 & 7 \end{bmatrix}$,verify that $(A - A^{\prime})$ is a skew-symmetric matrix.