JEE Main 2025 Jan 24 Shift 2, Mathematics Q19: Introduction To Probability

JEE Main2025Jan 24, Shift 2Mathematics

Let $A = [a_{ij}]$ 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 :

A
$\frac{5}{8}$
B
$\frac{3}{16}$
C
$\frac{1}{8}$
D
$\frac{3}{8}$

Solution

The sample space: each of the 4 entries can independently be $0$ or $1$ , so $∣ S ∣ = 2^{4} = 16$ .

A $2 \times 2$ matrix $(ab c d)$ is invertible iff $a d - b c \neq = 0$ . With entries in ${0, 1}$ , the only possibilities for $a d - b c$ are $- 1, 0, 1$ , so invertibility means $a d - b c = \pm 1$ .

Case $a d - b c = 1$ : $a d = 1$ and $b c = 0$ . $a d = 1 \Rightarrow a = d = 1$ ; $b c = 0$ gives 3 choices ( ${b, c} \subseteq {0, 1}$ minus the case $b = c = 1$ ): $(0, 0), (0, 1), (1, 0)$ . So $3$ matrices.

Case $a d - b c = - 1$ : $a d = 0$ and $b c = 1$ . $b c = 1 \Rightarrow b = c = 1$ ; $a d = 0$ gives 3 choices: $(0, 0), (0, 1), (1, 0)$ . So $3$ matrices.

Favourable cases $= 3 + 3 = 6$ .

$P (E) = \frac{6}{16} = \frac{3}{8}$

Correct option: (4)

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JEE Main 2025 Jan 24 Shift 2, Mathematics Q19: Introduction To Probability

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