Question

The sum of the values of $x$ so that the matrix $\begin{bmatrix}2 & 2 & 1 \\ 1 & 3 & 1 \\ 1 & 2 & 2\end{bmatrix}-x\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$ is singular, is

• A. 3
• B. 5
• C. 7
• D. 9

Answer 1

Answer: (A)

Let a matrix
$=\begin{bmatrix}2 & 2 & 1 \\ 1 & 3 & 1 \\ 1 & 2 & 2\end{bmatrix}-x\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}=\begin{bmatrix}2-x & 2 & 1 \\ 1 & 3-x & 1 \\ 1 & 2 & 2-x\end{bmatrix}$
$\because$ Matrix $A$ is a singular.
$\therefore |A|=0 \Rightarrow \begin{vmatrix}2-x & 2 & 1 \\ 1 & 3-x & 1 \\ 1 & 2 & 2-x\end{vmatrix}=0$
On applying $C_{1} \rightarrow C_{1}+C_{2}+C_{3}$, we get
$\begin{vmatrix}5-x & 2 & 1 \\ 5-x & 3-x & 1 \\ 5-x & 2 & 2-x\end{vmatrix}=0$
$\Rightarrow (5-x)\begin{vmatrix}1 & 2 & 1 \\ 1 & 3-x & 1 \\ 1 & 2 & 2-x\end{vmatrix}=0$
On applying $R_{2} \rightarrow R_{2}-1$ and $R_{3} \rightarrow R_{3}-R_{1}$, we get
$(5-x)\begin{vmatrix}1 & 2 & 1 \\ 0 & 1-x & 0 \\ 0 & 0 & 1-x\end{vmatrix}=0$
$\Rightarrow (5-x)(1-x)^{2}=0 $
$\Rightarrow x=1,1,5$
So, sum of the required values of $x$ is $7$ .