Let be a square matrix of order 2 with entries either 0 or 1. Let E be the event that A is an invertible matrix. Then the probability P(E) is: [2025]

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Solution

Q.
Let $A = [a_{i j}]$ be a square matrix of order 2 with entries either 0 or 1. Let E be the event that A is an invertible matrix. Then the probability P(E) is: [2025]

1 $\frac{5}{8}$

2 $\frac{3}{8}$

3 $\frac{3}{16}$

4 $\frac{1}{8}$

Ans.
(2)

Given : $A = {[a_{i j}]}_{2 \times 2}$ with entries 0 or 1.

Total possible ways for matrix A are $4 \times 4 = 16$ ways.

Here E = { $A | A$ is invertible}

$[\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}], [\begin{matrix} 1 & 1 \\ 1 & 0 \end{matrix}], [\begin{matrix} 1 & 0 \\ 1 & 1 \end{matrix}], [\begin{matrix} 0 & 1 \\ 1 & 1 \end{matrix}], [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] and [\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}]$ = 6 matrices

Thus, probability P(E) is given by $= \frac{6}{16} = \frac{3}{8}$ .

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