Let S = \ M = [ a _ ij ], a _ ij ∈ \ 0, 1, 2\ , \ 1 ≤ i, j ≤ 2\ \...

Given,

$S = \left\{M = \left[{a}_{ij}\right], {a}_{ij} \in \left\{0, 1, 2\right\}, \left\{1 \le i, j \le 2\right\}\right\}$

So, let M = \left[\begin{cases}a & b \\ c & d\end{cases}\right] where $a, b, c, d \in \left\{0, 1, 2\right\}$

So, total sample space will be ${3}^{4} = 81$

Now we will find $P\left(\overset{¯}{A}\right)$ or possibilities for $| M | = 0$

So, for $| M | = 0$ cases will be,

1.\left[$\begin{cases}1 & 1 \\ 1 & 1\end{cases}$\right]or \left[\begin{cases}0 & 0 \\ 0 & 0\end{cases}\right] or \left[$\begin{cases}2 & 2 \\ 2 & 2\end{cases}$\right] \to$Total matrix$= 3$<br />2. Two$1 ' s$and Two$0 ' s \to$Total matrix$= 4$<br />3. Two$2 ' s$and Two$0 ' s \to$Total matrix$= 4$<br />4. Two$1 ' s$and Two$2 ' s \to$Total matrix$= 4$<br />5. One$1$and three$0 ' s \to$Total matrix$= 4$<br />6. One$2$and Three$0 ' s \to$Total matrix$= 4$<br />7. One$1$and one$2$and two$0 ' s \to$Total matrix$= 8$</p><p>So, total cases for which \left|M\right| = 0$ is $31$,

So probability will be $P\left(\overset{¯}{A}\right) = \dfrac{31}{81}$
Hence, $P ( A ) = 1 - \dfrac{31}{81} = \dfrac{50}{81}$