Question

The sum of the values of $x$ so that the matrix $\left[\begin{array}{ccc}2& 2& 1\\ 1& 3& 1\\ 1& 2& 2\end{array}\right]-x\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$ is singular, is

• A. 3
• B. 5
• C. 7
• D. 9

Answer 1

Answer: (A)

Let a matrix
$=\left[\begin{array}{ccc}2& 2& 1\\ 1& 3& 1\\ 1& 2& 2\end{array}\right]-x\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}2-x& 2& 1\\ 1& 3-x& 1\\ 1& 2& 2-x\end{array}\right]$
$\because$ Matrix $A$ is a singular.
$\therefore |A|=0\Rightarrow \left|\begin{array}{ccc}2-x& 2& 1\\ 1& 3-x& 1\\ 1& 2& 2-x\end{array}\right|=0$
On applying $C_1\to C_1+C_2+C_3$, we get
$\left|\begin{array}{ccc}5-x& 2& 1\\ 5-x& 3-x& 1\\ 5-x& 2& 2-x\end{array}\right|=0$
$\Rightarrow (5-x)\left|\begin{array}{ccc}1& 2& 1\\ 1& 3-x& 1\\ 1& 2& 2-x\end{array}\right|=0$
On applying $R_2\to R_2-1$ and $R_3\to R_3-R_1$, we get
$(5-x)\left|\begin{array}{ccc}1& 2& 1\\ 0& 1-x& 0\\ 0& 0& 1-x\end{array}\right|=0$
$\Rightarrow (5-x)(1-x{)}^2=0$
$\Rightarrow x=1,1,5$
So, sum of the required values of $x$ is $7$ .