If a $2^{nd}$ order determinant with elements $0$ or $1$ is chosen from the set of all such determinants,find the probability that the determinant chosen is non-zero.

Match List-$I$ with List-$II$:

List-$I$	List-$II$
a) $A$ matrix which is not a square matrix	i) Symmetric matrix
b) $A$ square matrix $A' = A$	ii) Null matrix
c) The diagonal elements of a diagonal matrix are same	iii) Rectangular matrix
d) $A$ matrix which is both symmetric and skew symmetric	iv) Scalar matrix

Let $A = \begin{bmatrix} \cos \theta & 0 & -\sin \theta \\ 0 & 1 & 0 \\ \sin \theta & 0 & \cos \theta \end{bmatrix}$. If for some $\theta \in (0, \pi)$,$A^2 = A^T$,then the sum of the diagonal elements of the matrix $(A + I)^3 + (A - I)^3 - 6A$ is equal to . . . . . . .

Let $A$ be a matrix such that $A \cdot \begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix}$ is a scalar matrix and $|3A| = 108$. Then $A^2$ equals

If $A = \begin{bmatrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{bmatrix}$,prove that $A^{3} - 6A^{2} + 7A + 2I = 0$.

Suppose $A$ is a $3 \times 3$ matrix consisting of integer entries that are chosen at random from the set $\{-1000, -999, \ldots, 999, 1000\}$. Let $P$ be the probability that either $A^2 = -I$ or $A$ is diagonal,where $I$ is the $3 \times 3$ identity matrix. Then,