Search any question & find its solution
Question:
Answered & Verified by Expert
The sum of the distinct values of $x$ for which the matrix $A=\left[\begin{array}{lll}1 & 1 & x \\ 1 & x & 1 \\ x & 1 & 1\end{array}\right]$ has no inverse, is
Options:
Solution:
1444 Upvotes
Verified Answer
The correct answer is:
-1
If matrix $\mathrm{A}$ has no inverse, then $\operatorname{det}(\mathrm{A})=0$
$$
\begin{aligned}
& \Rightarrow\left|\begin{array}{ccc}
1 & 1 & x \\
1 & x & 1 \\
x & 1 & 1
\end{array}\right|=0 \\
& \Rightarrow x^3-3 x+2=0 \\
& \Rightarrow x^2(x-1)+x(x-1)-2(x-1)=0 \\
& \Rightarrow(x-1)(x+2)(x-1)=0 \\
& \Rightarrow(x-1)^2(x+2)=0 \Rightarrow x=-2,+1,+1
\end{aligned}
$$
Sum of distinct values of $x=-2+1=-1$.
$$
\begin{aligned}
& \Rightarrow\left|\begin{array}{ccc}
1 & 1 & x \\
1 & x & 1 \\
x & 1 & 1
\end{array}\right|=0 \\
& \Rightarrow x^3-3 x+2=0 \\
& \Rightarrow x^2(x-1)+x(x-1)-2(x-1)=0 \\
& \Rightarrow(x-1)(x+2)(x-1)=0 \\
& \Rightarrow(x-1)^2(x+2)=0 \Rightarrow x=-2,+1,+1
\end{aligned}
$$
Sum of distinct values of $x=-2+1=-1$.
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.