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 $P = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$ be a matrix. Three elements of this matrix $P$ are selected at random. $A$ is the event of having the three elements whose sum is odd. $B$ is the event of selecting the three elements which are in a row or column. Then $P(A) + P(A|B) =$?

If $A = \begin{bmatrix} 3 & 4 \\ 5 & 6 \end{bmatrix}$ and $B = \begin{bmatrix} x & 0 \\ 0 & y \end{bmatrix}$,where $x, y \in \mathbb{N}$,then:

Let $M$ denote the set of all real matrices of order $3 \times 3$ and let $S=\{-3,-2,-1,1,2\}$. Let $S_1=\{A=[a_{ij}] \in M: A=A^{T} \text{ and } a_{ij} \in S, \forall i, j\}$,$S_2=\{A=[a_{ij}] \in M: A=-A^{T} \text{ and } a_{ij} \in S, \forall i, j\}$,and $S_3=\{A=[a_{ij}] \in M: a_{11}+a_{22}+a_{33}=0 \text{ and } a_{ij} \in S, \forall i, j\}$. If $n(S_1 \cup S_2 \cup S_3)=125 \alpha$,then $\alpha$ equals.

Which of the following statements is correct about two square matrices $A$ and $B$ of the same order $n$?

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