JEE Advanced 2023 · Paper 2, Mathematics question 12

Determinant. Expand along the third row:
$$\det=-5\begin{vmatrix}a&b\\c&d\end{vmatrix}=-5(ad-bc).$$
So the matrix is invertible iff $ad\ne bc$. There are $8^4=4096$ matrices in all.

Count $ad=bc$. The set is $\{0\}$ together with seven primes.

• $ad=0$: $15$ ordered pairs $(a,d)$, and likewise 15 pairs $(b,c)$ with $bc=0$. That gives $15^2=225$.
• $ad\ne0$: by unique factorisation, $\{b,c\}=\{a,d\}$.
  • $a=d$: 1 way for $(b,c)$, over 7 pairs: 7.
  • $a\ne d$: 2 ways, over 42 pairs: 84.

Total singular: $225+7+84=316$.

$$4096-316=3780.$$