If a matrix is chosen at random from the set of all $3 \times 3$ non-zero matrices whose entries are the elements of the set $\{-1, 0, 1\}$, then the probability that the matrix is skew-symmetric is

Let $A$ be a $2 \times 2$ matrix with real entries. Let $I$ be the $2 \times 2$ identity matrix. Denote by $tr(A)$ the sum of diagonal entries of $A$. Assume that $A^2 = I$.
Statement-$1$: If $A \neq I$ and $A \neq -I$,then $\det(A) = -1$.
Statement-$2$: If $A \neq I$ and $A \neq -I$,then $tr(A) \neq 0$.

If $A+B=\left[\begin{array}{cr}1 & \tan \frac{\theta}{2} \\ -\tan \frac{\theta}{2} & 1\end{array}\right]$ where $A$ is a symmetric matrix and $B$ is a skew-symmetric matrix,then the matrix $\left(A^{-1} B+A B^{-1}\right)$ at $\theta=\frac{\pi}{6}$ is given by

Let $A$ be a non-zero periodic matrix with period $4$ and $A^{12} + B = I$,where $I$ is the identity matrix and $B$ is any square matrix of the same order as $A$. The matrix product $AB$ is equal to:

Consider the matrix $P = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}$. Let the transpose of a matrix $X$ be denoted by $X^T$. Then the number of $3 \times 3$ invertible matrices $Q$ with integer entries,such that $Q^{-1} = Q^T$ and $PQ = QP$ is

Let $A$ and $B$ be $3 \times 3$ real matrices such that $A$ is a symmetric matrix and $B$ is a skew-symmetric matrix. Then the system of linear equations $(A^{2}B^{2} - B^{2}A^{2})X = O$,where $X$ is a $3 \times 1$ column matrix of unknown variables and $O$ is a $3 \times 1$ null matrix,has ....... .