JEE Main 2025 — 7 April, Morning Shift — Question 42

Let $A=\left[\begin{array}{ll}a & b\\ c & d\end{array}\right]$

for $A$ to be singular matrix

$a d=b c$

Case 1 : exactly 1 number is used $\Rightarrow{ }^{4} C_{1}$ ways

Case 2: exactly 2 numbers is used $\Rightarrow{ }^{4} C_{2}$ ways

Case 3: exactly 3 numbers used $\Rightarrow$ none will be singular.

Case 4: exactly 4 numbers is used

$\Rightarrow a b=c d \Rightarrow 2 \times 9=3 \times 6$

$\Rightarrow{ }^{4} C_{1} \times 2!=8$ matrix.

$\therefore$ Total ways $\Rightarrow 4+6 \times 4+8=36$ matrices.