If $A=\left[\begin{array}{ccc}x & 1 & 2 \ 2 & 4 & x \ -3 & 3 & 2\end{array} ight]$ is a singular matrix and the distinct values of $x$ are $x_1$ and $x_2$, then $x_1+x_2+x_1 x_2=$.

Question

If $A=\left[\begin{array}{ccc}x & 1 & 2 \ 2 & 4 & x \ -3 & 3 & 2\end{array}ight]$ is a singular matrix and the distinct values of $x$ are $x_1$ and $x_2$, then $x_1+x_2+x_1 x_2=$.

Aicharya · Accepted Answer

Given, $A=\left[\begin{array}{ccc}x & 1 & 2 \ 2 & 4 & x \ -3 & 3 & 2\end{array}ight]$ is singular, when $ \begin{aligned} & \left|\begin{array}{lll} A \mid=0 \ x & 1 & 2 \ 2 & 4 & x \ -3 & 3 & 2 \end{array}ight|=0 \ & \Rightarrow \quad x(8-3 x)-1(4+3 x)+2(6+12)=0 \ & \Rightarrow \quad 8 x-3 x^2-4-3 x+36=0 \ & \Rightarrow \quad \quad 3 x^2-5 x-32=0 \ & 	herefore x_1+x_2=\frac{5}{3} 	ext { and } x_1 x_2=-\frac{32}{3} \ & x_1+x_2+x_1 x_2=\frac{5}{3}-\frac{32}{3}=-\frac{27}{3}=-9 \end{aligned} $

If $A=\left[\begin{array}{ccc}x & 1 & 2 \\ 2 & 4 & x \\ -3 & 3 & 2\end{array}\right]$ is a singular matrix…

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