If $A$ and $B$ are square matrices of the same order and $B$ is a skew-symmetric matrix,then $A^{\prime} B A$ is

The trace of the matrix $A = \begin{bmatrix} 0 & 7 & 9 \\ 11 & 8 & 9 \end{bmatrix}$ is defined only for square matrices. If we consider the matrix $A = \begin{bmatrix} 1 & -5 & 7 \\ 0 & 7 & 9 \\ 11 & 8 & 9 \end{bmatrix}$,what is its trace?

The matrix $A = \begin{bmatrix} 1/\sqrt{2} & 1/\sqrt{2} \\ -1/\sqrt{2} & -1/\sqrt{2} \end{bmatrix}$ is

If for the matrix $A = \begin{bmatrix} 1 & -\alpha \\ \alpha & \beta \end{bmatrix}$,$AA^{T} = I_{2}$,then the value of $\alpha^{4} + \beta^{4}$ is ....... .

$P$ is an orthogonal matrix and $A$ is a periodic matrix with period $4$,and $Q = PAP^T$. Then $X = P^TQ^{2005}P$ will be equal to

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