Let R = { ( a 3 b c 2 d 0 5 0 ) : a , b , c , d { 0 , 3 , 5 , 7 , 11 , 13 , 17 , 19 } } . Then the number of invertible matrices in R is [2023]

Question

Let $R = {(\begin{matrix} a & 3 & b \\ c & 2 & d \\ 0 & 5 & 0 \end{matrix}) : a, b, c, d \in {0, 3, 5, 7, 11, 13, 17, 19}} .$ Then the number of invertible matrices in $R$ is [2023]

Smart Achievers · Accepted Answer

(3780)

Let us calculate when $| R |$ $= 0$

Case I: $a d = b c = 0$

If $a d = 0$ (when none of $a$ and $d$ is 0),

$Total = 8^{2} - 1 = 15 ways$

Similarly, $b c = 0 \Rightarrow 15 ways$

$∴$ $15 \times 15 = 225$ ways for $a d = b c = 0$

Case II: $a d = b c \neq 0$

Either $a = d = b = c$ or $a \neq d,$ $b \neq c$ but $a d = b c$

${}^{7}C_{1} = 7 ways$ or ${}^{7}C_{2} \times 2 \times 2 = 84 ways$

$Total = 7 + 84 = 91 ways$

$∴$ $| R |$ $= 0$ in $225 + 91 = 316$ ways

$| R |$ $\neq 0$ in $8^{4} - 316 = 3780$

Solution

Q.
Let $R = {(\begin{matrix} a & 3 & b \\ c & 2 & d \\ 0 & 5 & 0 \end{matrix}) : a, b, c, d \in {0, 3, 5, 7, 11, 13, 17, 19}} .$ Then the number of invertible matrices in $R$ is [2023]

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Solution

Q. Let R={(a3bc2d050): a,b,c,d∈{0,3,5,7,11,13,17,19}}. Then the number of invertible matrices in R is [2023]

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Q.
Let $R = {(\begin{matrix} a & 3 & b \\ c & 2 & d \\ 0 & 5 & 0 \end{matrix}) : a, b, c, d \in {0, 3, 5, 7, 11, 13, 17, 19}} .$ Then the number of invertible matrices in $R$ is [2023]