Question

If $A=\left[\begin{array}{ccc}3& -1+x& 2\\ 3& -1& x+2\\ x+3& -1& 2\end{array}\right]$ is singular matrix and $x\in [-5,-2]$ then x=?$

• A. $0$
• B. $-2$
• C. $-4$
• D. $0,-4$

Answer 1

Answer: (C)

$-4$

Given $A=\left[\begin{array}{ccc}3& -1+x& 2\\ 3& -1& x+2\\ x+3& -1& 2\end{array}\right]$ is a singular matrix.

$\therefore |A|=0$

$\left|\begin{array}{ccc}3& -1+x& 2\\ 3& -1& x+2\\ x+3& -1& 2\end{array}\right|=0$

$\Rightarrow 3(-2+x+2)-(-1+x)(6-(x+3\left)\right(x+2\left)\right)+2(-3+x+3)=0$

$\Rightarrow 3x-(x-1)(6-x^2-5x-6)+2x=0$

$\Rightarrow 5x-(x-1)(-x^2-5x)=0$

$\Rightarrow 5x-(-x^3-5x^2+x^2+5x)=0$

$\Rightarrow 5x+x^3+4x^2-5x=0$

$\Rightarrow x^3+4x^2=0$

$\Rightarrow x^2(x+4)=0$

$\Rightarrow x=0andx=-4$

It is given that $x\in [-5,-2]$

$\therefore x=-4$