Let $R=\left\{\left(\begin{array}{lll}a & 3 & b \ c & 2 & d \ 0 & 5 & 0\end{array} ight): a, b, c

Question

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

Professor PC Thomas Classes · Answer

(3780) : $\Delta=\left|\begin{array}{lll}a & 3 & b \ c & 2 & d \ 0 & 5 & 0\end{array}ight|=-5(a d-b c)$ Number of invertible matrices $=$ Total numer of matrices - Total number of non invertible matrices ( $\Delta=0$ )\ Total number of matrices $=8 	imes 8 	imes 8 	imes 8$ $=4096$ Now, $\Delta=0 \Rightarrow a d=b c$\ Case (1): (i) When all four $a, b, c, d$ are zero $=1$\ (ii) When three zero, one non-zero $=7 	imes \frac{4!}{3!}=28$\ (iii) When two zero and two non zero\ $={ }^{2} C_{1} \cdot{ }^{2} C_{1} \cdot 7 \cdot 7=196$\ Case (2): When all 4 are non zero\ (i) All 4 are same $=7$\ (ii) 2same +2 other same $={ }^{7} C_{2} \cdot{ }^{2} C_{1} \cdot{ }^{2} C_{1}$ $=21 	imes 4=84$\ Total non invertible matrices $=1+28+196$ $+7+84=316$\ Number of invertible matrices $=4096-316$ $=3780$\