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

Given,

$R=\left[\begin{array}{lll}a& 3& b\\ c& 2& d\\ 0& 5& 0\end{array}\right]$

$\Rightarrow \left|R\right|=-5\left|\begin{array}{lc}a& b\\ c& d\end{array}\right|$

We know that, for invertible matrices $\left|R\right|\ne 0$

Now $\left|R\right|$ 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 $a$)

$\to 4\times 7^2\text{ ways }=196$

(ii) Any three of $a, b, c, d$ are zeroes

$\to {}^{4}C_3\times 7=28$

(iii) All four of $a, b, c, d$ are zeroes

$\to 1$

(iv) All four of $a, b, c, d$ are non-zero but same number

$\to 7$

(v) When two are alike and $2$ other are alike (non-zero)

$\to 7{\mathrm{C}}_2\times 2\times 2=84$

Number of invertible matrices

$=8^4-196-28-1-7-84=3780$