Question

The matrix $\left|\begin{array}{ccc}5& 10& 3\\ -2& -4& 6\\ -1& -2& b\end{array}\right|$ is a singular matrix, if is equal to:

• A. $-3$
• B. $3$
• C. $0$
• D. for any value of $b$

Answer 1

Answer: (D)

If any square matrix is singular, then the value of determinant is zero.
Let $A=\left|\begin{array}{ccc}5& 10& 3\\ 2& -4& 6\\ -1& -2& b\end{array}\right|$
Since, $A$ is singular.
$\therefore \left|\begin{array}{ccc}5& 10& 3\\ -2& -4& 6\\ -1& -2& b\end{array}\right|=0$
$\Rightarrow 5(-4b+12)-10(-2b+6)+3(4-4)=0$
$\Rightarrow -20b+60+20b-60+0=0$
$\Rightarrow 0=0$
$\therefore$ The given matrix is singular for any value of $b$ .
For a non-singular matrix, the value of determinant is non-zero.