JEE Main 2022 — Matrices Question with Solution

Let $\mathrm{A}=\left[\begin{array}{ll}\mathrm{a}& \mathrm{b}\\ \mathrm{c}& \mathrm{d}\end{array}\right]$ where $\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}$ are prime.
Total number of matrices A formed $=10^4$
Let $\mathrm{S}$ be a matrix from set of matrices $\mathrm{A}$ such that $\mathrm{S}$ is singular.
For singular matrix $|\mathrm{S}|=\mathrm{ad}-\mathrm{bc}=0\Rightarrow \mathrm{ad}=\mathrm{bc}$
Number of singular matrices $=$ All entries are same $+$ only two prime number are used in matrix
then number of matrices with two prime numbers $=(10\times 9)\times 2!=180$
then number of matrices with one prime number $=10\times 1=10$
So required probability $=\frac{10\times 9\times 2!+10}{10^4}=\frac{19}{1000}=0.019$