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Let $\quad S =\left\{ M =\left[ a _{ ij }\right], a _{ ij } \in\{0,1,2\}, 1 \leq i , j \leq 2\right\}$ be a sample space and $A=\{M \in S: M$ is invertible $\}$ be an event. Then $P ( A )$ is equal to
$\frac{50}{81}$
$\frac{47}{81}$
$\frac{49}{81}$
$\frac{16}{27}$
Solution
$M\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]$, where $a , b , c , d , \in\{0,1,2\}$
$n(s)=3^4=81$
we first bound $p (\overline{ A })$
$| m |=0 \Rightarrow ad = bc$
$ad = bc =0 \Rightarrow \text { no. of }(a, b, c, d)=\left(3^2-2^2\right)^2=25$
$ad = bc =1 \Rightarrow \text { no. of }( a , b , c , d )=1^2=1$
$ad = bc =2 \Rightarrow \text { no. of }( a , b , c , d )=2^2=4$
$ad = bc =4 \Rightarrow \text { no. of }( a , b , c , d )=1^2=1$
$: P (\overline{ A })=\frac{31}{81} \Rightarrow p ( A )=\frac{50}{81}$
