If A, B and C are three singular matrices given by $A=\left[\begin{array}{cc}1 & 4 \\ 3 & 2 a\end{array}\right], \quad B=\left[\begin{array}{cc}3 b & 5 \\ a & 2\end{array}\right]$ and $C=\left[\begin{array}{cc}a+b+c & c+1 \\ a+c & c\end{array}\right]$, then the value of $a b c$ is:

45

The correct answer is Option (3) → 45

For A, B and C to be non-singular. |A|, |B| and |C| should be equal to zero.

$⇒|A|=2a-12=0$

$⇒a=\frac{12}{2}=6$

and,

$|B|=6b-5a$

$⇒6b-5a=0$

$⇒b=\frac{5×6}{6}=5$

and,

$|C|=c(a+b+c)-(a+c)(c+1)=0$

$⇒c(11+c)-(6+c)(c+1)=0$

$⇒11c+c^2-6-c^2-7c=0$

$⇒4c=6$

$⇒c=\frac{3}{2}$

$a=6,b=5,c=\frac{3}{2}$

$∴abc=6×5×\frac{3}{2}=9×5=45$