Q35 / 75 · JEE Main 2025 (24 Jan Shift 2) · Mathematics
Let $A=\left[a_{i j}\right]$ 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 $\mathrm{P}(\mathrm{E})$ is :
- A$\frac{3}{16}$
- B$\frac{5}{8}$
- Correct answerC$\frac{3}{8}$
- D$\frac{1}{8}$
Solution
$A=\left[a_y\right]_{2 \times 2}$ and entries are 0 or 1.
$\begin{aligned}
& \therefore\left|\begin{array}{ll}
a & b \\
c & d
\end{array}\right|=0 \\
& \Rightarrow a d-b c=0
\end{aligned}$
Case I: $a d=b c=1$
$\therefore \quad a=b=c=d=1$
Case II: $a d=b c=0$
$\begin{array}{ll}
a=0, d=0 & b=0, c=0 \\
a=0, d=1 & b=0, c=1 \\
a=1, d=0 & b=1, c=0
\end{array}$
$\therefore$ Total 10 cases when matrix is non invertible Total possible matrix $=2^4=16$
Required probability of invertible
$=\frac{16-10}{16}=\frac{6}{16}=\frac{3}{8}$
Hence, the correct answer is option: 'C'.