Question

The value of $x$, for which the matrix $A=\left[\begin{array}{cc}{e}^{x-2}& e^{7+x}\\ e^{2+x}& e^{2x+3}\end{array}\right]$ is singular, is

• A. $9$
• B. $8$
• C. $7$
• D. $6$
• E. $5$

Answer 1

The correct option is B $8$

Given, $A=\left[\begin{array}{cc}{e}^{x-2}& e^{7+x}\\ e^{2+x}& e^{2x+3}\end{array}\right]$

Since $\text{Matrix}A$ is singular,
Therefore, $|A|=0$

$\Rightarrow \left[\begin{array}{cc}{e}^{x-2}& e^{7+x}\\ e^{2+x}& e^{2x+3}\end{array}\right]=0$

$\Rightarrow e^{(x-2+2x+3)}-e^{(2+x+7+x)}=0$

$\Rightarrow e^{(3x-1)}-e^{(2x+9)}=0$

$\Rightarrow e^{3x+1}=e^{2x+9}$

On comparing, we get
$3x+1=2x+9$

$\Rightarrow x=8$.