Question

The number of value of $x$ in the closed interval $[-4,-1]$, the matrix $\left[\begin{array}{ccc}3& -1+x& 2\\ 3& -1& x+2\\ x+3& -1& 2\end{array}\right]$ is singular is

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is C $2$

For a singular matrix the value of the determinant = 0

$\left[3\right(-1\left)\right(2)+(-1+x\left)\right(x+2\left)\right(x+3)+(2\left)\right(3\left)\right(-1\left)\right]$

$\times [-(2\left)\right(-1\left)\right(x+3)-(-1+x\left)\right(3\left)\right(2)-(3\left)\right(x+2\left)\right(x+3\left)\right]=0$

$-6+x^3+4x^2+x-6-6+2x+6+6-6x-3x^2-15x-18=0$

$x^3+x^2-18x-24=0$

$x=-4$

$x=(3-\sqrt{33})/2=-1.372$

$x=(3+\sqrt{33})/2=4.372$

Value of x in the closed interval $[-4,-1]$ are $-4,-1.372$