Let R= ( lll a & 3 & ~b c & 2 & ~d 0 & 5 & 0 ): a, b, c, d 0,3,5,… - Vidyadip

MCQ

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

A
$500$
✓
$3780$
C
$515$
D
$520$

✓

Answer

Correct option: B.

$3780$

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

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

Now $\mathrm{ad}=0$

$\Rightarrow$ Total - (When none of a & $d$ is 0 )

$=8^2-1=15$ ways

Similarly bc $=0 \Rightarrow 15$ ways

$\therefore 15 \times 15=225$ ways of $a d=b c=0$

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

either $a=d=b=c \quad$ OR $\quad a \neq d, b \neq d$ but $a d=b c$

${ }^7 \mathrm{C}_1=7$ ways

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

Total 91 ways

$\therefore|\mathbb{R}|=0 \text { in } 225+91=316 \text { ways }$

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

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