Let S = { M = [ a i j ] , a i j { 0 , 1 , 2 } , 1 i , j 2 } be a sample space and A = { M S : M is invertible } be an event. Then P ( A ) is equal to [2023]

Question

Let $S = {M = [a_{i j}], a_{i j} \in {0, 1, 2}, 1 \leq i, j \leq 2}$ be a sample space and $A = {M \in S : M is invertible}$ be an event. Then $P (A)$ is equal to [2023]

Smart Achievers · Accepted Answer

(1)

$Let M = [\begin{matrix} p & q \\ r & s \end{matrix}]$ $where p, q, r, s \in {0, 1, 2}$

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

$First, we find p (\bar{A})$

$\bar{A} = {M \in S : | M | = 0}$

$| M |$ $= 0 \Rightarrow p s = q r$

$p s = q r = 0 \Rightarrow number of (p, q, r, s) = {(3^{2} - 2^{2})}^{2} = 25$

$p s = q r = 1 \Rightarrow number of (p, q, r, s) = 1^{2} = 1$

$p s = q r = 2 \Rightarrow number of (p, q, r, s) = 2^{2} = 4$

$p s = q r = 4 \Rightarrow number of (p, q, r, s) = 1^{2} = 1$

$P (\bar{A}) = \frac{31}{81} \Rightarrow P (A) = \frac{50}{81}$

Solution

Q.
Let $S = {M = [a_{i j}], a_{i j} \in {0, 1, 2}, 1 \leq i, j \leq 2}$ be a sample space and $A = {M \in S : M is invertible}$ be an event. Then $P (A)$ is equal to [2023]

1 $\frac{50}{81}$

2 $\frac{47}{81}$

3 $\frac{49}{81}$

4 $\frac{16}{27}$

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Solution

Q. Let S={M=[aij],aij∈{0,1,2},1≤i,j≤2} be a sample space and A={M∈S:M is invertible} be an event. Then P(A) is equal to [2023]

1 5081

2 4781

3 4981

4 1627