Let S=M=[a_ij], a_ij0,1,2, 1 i,j2 and A=M S: M invertible.

Question

Let $S=\{M=[a_{ij}], a_{ij}\in\{0,1,2\}, 1\le i,j\le2\}$ and $A=\{M\in S: M	ext{ invertible}\}$. Find $P(A)$.

Accepted Answer

$n(S)=3^4=81$. For $M=\begin{pmatrix}a&b\c&d\end{pmatrix}$ singular, $ad=bc$. Counting: $ad=bc=0$ gives 25 combos; $ad=bc=1$ gives 1; $ad=bc=2$ gives 4; $ad=bc=4$ gives 1. Total singular $=31$.

$$P(A) = 1-\frac{31}{81}=\frac{50}{81}$$

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