Question

The value of $x$ for which the matrix $A=\left[\begin{array}{ccc}x-1& 1& 1\\ 1& x-1& 1\\ 1& 1& x-1\end{array}\right]$
Is non-singular

• A. $x\in R-\{-1,2\}$
• B. $x\in R-\{1,2\}$
• C. $x\in R-\{-2,2\}$
• D. $Noneofthese$

Answer 1

The correct option is A $x\in R-\{-1,2\}$

$A=\left[\begin{array}{ccc}x-1& 1& 1\\ 1& x-1& 1\\ 1& 1& x-1\end{array}\right]$

matrix ${}^{\prime }{A}^{\prime }$ is non-singular

$\therefore |A|\ne 0$

$=\left|\begin{array}{ccc}x-1& 1& 1\\ 1& x-1& 1\\ 1& 1& x-1\end{array}\right|\ne 0$

$\Rightarrow (x-1)[x^2+1-2x-1]-1(x-2)+1(1-x+1)\ne 0$

$\Rightarrow (x-1).x(x-2)+2(2-x)\ne 0$

$\Rightarrow (x-1)x(x-2)-2(x-2)\ne 0$

$\Rightarrow (x-2)(x^2-2x+x-2)\ne 0$

$\Rightarrow (x-2)\left(x\right(x-2)+1(x-2\left)\right)\ne 0$

$\Rightarrow (x-2{)}^2(x+1)\ne 0$

$\therefore x\ne 2,x\ne -1$

$\therefore x\in R-\{-1,2\}$