Question:
The matrix $\left[\begin{array}{ccc}5 & 10 & 3 \\ -2 & -4 & 6 \\ -1 & -2 & b\end{array}\right]$ is a singular matrix, if the value of $b$ is
(a) $-3$
(b) 3
(c) 0
(d) non-existent
Solution:
(d) non-existent
For any singular matrix, the value of the determinant is 0 .
Here,
$A=\left[\begin{array}{ccc}5 & 10 & 3 \\ -2 & -4 & 6 \\ -1 & -2 & b\end{array}\right]$
$|A|=5(-4 b+12)-10(-2 b+6)+3(4-4)=0$
$\Rightarrow-20 b+60+20 b-12=0$
Hence, $b$ is non-existent.
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