JEE Advanced 2023 — Paper 2 — Question 12

Total matrices $=8^{4}=4096$ $|\mathrm{R}|=5(\mathrm{bc}-\mathrm{ad})$ No. of non-invertible matrices:

$$\mathrm{bc}=\mathrm{ad}$$

Case I: if a, b, c, $d \neq 0$, then cases $={ }^{7} C_{2} \cdot 2!2!+{ }^{7} C_{1}(1)=91$ Case II: if ad = bc $=0$, then cases $={ }^{15} \mathrm{C}_{1} \cdot{ }^{15} \mathrm{C}_{1}=225$ (ad $\&$ bc can take any combination from $0 \times 0,0 \times 3,0 \times 5, \ldots, 0 \times 19,3 \times 0,5 \times 0, \ldots 19 \times 0$ ) No. of invertible matrices $=4096-(91+225)=3780$