Question

The roots of the equation $\left|\begin{array}{ccc}x-1& 1& 1\\ 1& x-1& 1\\ 1& 1& x-1\end{array}\right|=0$ are

• A. $1,2$
• B. $-1,2$
• C. $1,-2$
• D. $-1,-2$

Answer 1

Answer: (B)

$-1,2$

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

Doing Column operation $C_1\to C_1+C_2,C_3\to C_3+C_2$

Determinant $D=\left|\begin{array}{ccc}x& 1& 2\\ x& x-1& x\\ 2& 1& x\end{array}\right|$

Doing Column operation $C_1\to C_1-C_3$ and taking out $(x-2)$ $D=(x-2)\left|\begin{array}{ccc}1& 1& 2\\ 0& x-1& x\\ -1& 1& x\end{array}\right|$

Doing column operation $C_2\to C_2+C_1$

$D=(x-2)\left|\begin{array}{ccc}1& 2& 2\\ 0& x-1& x\\ -1& 0& x\end{array}\right|$

By expanding the Determinant

$D=(x-2)\left[\right(x-1)x-(2x-2(x-1)\left)\right]=(x-2)[x^2-x-2]$

$D=(x-2{)}^2(x+1)$

Roots of the equation are $x=2,-1$