Let $S=\{M=[a_{ij}], a_{ij} \in\{0,1,2\}, 1 \leq i, j \leq 2\}$ be a sample space and $A=\{M \in S: M\text{ is invertible}\}$ be an event. Then $P(A)$ is equal to

JEE Main · 2023 · 11 Apr 2023 · Shift 1 · Probability

Let

S = {M = [a_{ij}], a_{ij} \in {0, 1, 2}, 1 \leq i, j \leq 2}

be a sample space and

A = {M \in S : M is invertible}

be an event. Then

P (A)

is equal to

Answer: (A)

Total

2 \times 2

matrices with entries from

{0, 1, 2}

n (S) = 3^{4} = 81

. A matrix

[a c b d]

is non-invertible if

a d = b c

. Count cases:

a d = b c = 0

gives 25 matrices;

a d = b c = 1

gives 1;

a d = b c = 2

gives 4;

a d = b c = 4

gives 1. Total non-invertible = 31. So

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

JEE Main · 2023 · 11 Apr 2023 · Shift 1 · Single correct option