Let R = {({a 3 b} {c 2 b} {0 5 0}) : a, b, c, d ∈ {0.,3,5,7,11,13,17,19}}.

Question

Let R = \(\left\{ \begin{pmatrix} a&3&b\\c&2&d\\0&5&0\end{pmatrix}:a, b,c,d \in \{0,3,5,7,11,13,17,19\}\right\}\). Then the number of invertible matrices in R is ____.

Answer 1

\(|R| = - 5\begin{vmatrix}a&b\\c&d \end{vmatrix}\)

|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