Question

If rank of $\left(\begin{array}{ccc}x& x& x\\ x& x^2& x\\ x& x& x+1\end{array}\right)$ is $1$, then

• A. $x=0$ or $x=1$
• B. $x=1$
• C. $x=0$
• D. $x\ne 0$

Answer 1

The correct option is C $x=0$

Given matrix is $\left(\begin{array}{ccc}x& x& x\\ x& x^2& x\\ x& x& x+1\end{array}\right)$

Now performing the operations of : $R_1\to R_1-R_3$ and $R_2\to R_2/R_1$

we have : $\left(\begin{array}{ccc}0& 0& 1\\ 1& x& 1\\ x& x& x+1\end{array}\right)$

For the rank of the matrix to be $1$, $2$ rows of the above matrix should be same:

If we put $x=0$, we have the matrix become

$\Rightarrow \left(\begin{array}{ccc}0& 0& 1\\ 1& 0& 1\\ 0& 0& 1\end{array}\right)$

Thus this justifies the given condition that the rank of the matrix is $1$.

Thus we have $x=0$.