If $A$ and $B$ are $3 \times 3$ order matrices and $|A|=5$,$|B|=3$,then $|3AB|=$ . . . . . . .

If $A$ is a skew-symmetric matrix of order $3$ and $X$ is another matrix of the same order,then $|XA + AX^T|$ is (where $|P|$ denotes the determinant of matrix $P$).

If $A$ is a skew-symmetric matrix of order $n$,and $C$ is a column matrix of order $n \times 1$,then $C^T AC$ is

If $A = \begin{bmatrix} -1 & x & -3 \\ 2 & 4 & z \\ y & 5 & -6 \end{bmatrix}$ is a symmetric matrix and $B = \begin{bmatrix} 0 & 2 & q \\ p & 0 & -4 \\ -3 & r & s \end{bmatrix}$ is a skew-symmetric matrix,then $|A| + |B| - |AB| = $

Let $M$ be a $2 \times 2$ symmetric matrix with integer entries. Then $M$ is invertible if

If $P$ and $Q$ are square matrices such that $P^{2006} = O$ and $PQ = P + Q$,then $\det(Q)$ will be