JEE Main · 2025 · 07 Apr 2025 · Shift 1 · Matrices

A matrix $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ with entries drawn from $\{2,3,6,9\}$ is singular when $ad=bc$. Grouping the ordered pairs of entries by their product, the number of pairs $(x,y)$ with $xy=p$ is one for each of $p=4,9,36,81$, two for each of $p=6,12,27,54$, and four for $p=18$ (namely $2\cdot 9,9\cdot 2,3\cdot 6,6\cdot 3$). A singular matrix is built by choosing $(a,d)$ and $(b,c)$ with the same product, so the count is ${\sum }_{p}N(p{)}^2=4\cdot 1^2+4\cdot 2^2+1\cdot 4^2=4+16+16=36$. Hence there are $36$ singular matrices.