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

$\text{Given: } M = \begin{pmatrix} a & 3 & b \\ c & 2 & d \\ 0 & 5 & 0 \end{pmatrix},\quad a,b,c,d \in \{0,3,5,7,11,13,17,19\}$ $\text{To find: the number of invertible matrices in } R$ $1.\ \text{invertible} \iff \det M \neq 0$ $2.\ \text{compute } \det M$ $3.\ \text{count total, subtract the non-invertible}$ $\text{invertible} \iff \det M \neq 0$ $\det M = 0\cdot C_{31} + 5\cdot C_{32} + 0\cdot C_{33} = 5\,C_{32}$ $C_{32} = (-1)^{3+2}M_{32} = -\begin{vmatrix} a & b \\ c & d \end{vmatrix}$ $\det M = -5(ad - bc) = 5(bc - ad)$ $\det M = 0 \iff bc = ad$ $\text{total matrices} = 8^4 = 4096$ $\text{Case 1: } ad = 0,\ bc = 0$ $\text{Case 2: } a,d \neq 0 \text{ and } bc = ad$ $ad=0:\ 8^2-7^2 = 15,\qquad bc=0:\ 15$ $15 \times 15 = 225$ $a,d \in \{3,5,7,11,13,17,19\}\ \ (7 \text{ primes})$ $a \neq d:\ 42 \times 2 = 84;\qquad a = d:\ 7 \times 1 = 7$ $84 + 7 = 91$ $\text{non-invertible} = 225 + 91 = 316$ $\text{invertible} = 4096 - 316 = 3780$ $\boxed{3780}$