JEE Advanced Mathematics Question (2023) — Solution

Step-by-step solution

Given, R = a 3 b c 2 d 0 5 0 ⇒ | R | = - 5 a b c d We know that, for invertible matrices  R ≠ 0 Now  | 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 × 7 2  ways  = 196 (ii) Any three of a ,   b ,   c ,   d  are zeroes → C 3 4 × 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) → 7 C 2 × 2 × 2 = 84 Number of invertible matrices = 8 4 - 196 - 28 - 1 - 7 - 84 = 3780