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

Total =$8^4=4096$
$\left|\mathrm{M}\right|\Rightarrow \mathrm{ad}=\mathrm{bc}$
 $$\begin{gathered} \text{ (i) Two of }a,b,c\text{, }d\text{ are zeroes which can be ( }a\text{ and }b)\text{, ( }b\text{ and }d\text{ ), ( }d\text{ and }c\text{ ) or (c and }a\text{ ) } \\ \to 4\times 7^2\text{ ways }=196 \end{gathered}$$

$$\begin{gathered} \text{ (ii) Any three of a, b, c, }d\text{ are zeroes } \\ {\to }^4{C}_3\times 7=28 \end{gathered}$$$$\begin{gathered} \text{ (v) When two are alike and }2\text{ other are alike (non-zero) } \\ {\to }^7{\mathrm{C}}_2\times 2\times 2=84 \end{gathered}$$

$All 4 a,b,c,d are zeros\to 1$

$\text{ (iv) All four of a, b, c, }d\text{ are non-zero but same number→7 }$

$\text{ Number of invertible matrices }=8^4-196-28-1-7-84=3780$