\(|R| = - 5\begin{vmatrix}a&b\\c&d \end{vmatrix}\)
|R| can be zero in following cases:
(i) Two of a, b, c, d are zeroes which can be (a and b), (b and d), (d and c) or (c and a) → 4 × 72 ways = 196
(ii) Any three of a, b, c, d are zeroes
→ 4C3 × 7 = 28
(iii) All four of a, b, c, d are zeroes
→ 1
(iv) All four of a, b, c, d are non-zero but same number
→ 7
(v) When two are alike and 2 other are alike (non-zero)
→ 7C2 × 2 × 2 = 84
Number of invertible matrices = 84 – 196 – 28 – 1 – 7– 84 = 3780