Question

If $n$ is a positive integer, then the value of the determinant $\left|\begin{array}{ccc}{a}^n-x& a^{n+1}-x& a^{n+2}-x\\ a^{n+3}-x& a^{n+4}-x& a^{n+5}-x\\ a^{n+6}-x& a^{n+7}-x& a^{n+8}-x\end{array}\right|=$

• A. $1$
• B. $0$
• C. $-1$
• D. None of these

Answer 1

The correct option is B $0$

Given

$A=\left|\begin{array}{ccc}{a}^n-x& a^{n+1}-x& a^{n+2}-x\\ a^{n+3}-x& a^{n+4}-x& a^{n+5}-x\\ a^{n+6}-x& a^{n+7}-x& a^{n+8}-x\end{array}\right|$

Applying $c_3\to c_3-a{c}_2,c_2\to c_2-a{c}_1$ we get

$A=\left|\begin{array}{ccc}{a}^n-x& (a-1)x& (a-1)x\\ a^{n+3}-x& (a-1)x& (a-1)x\\ a^{n+6}-x& (a-1)x& (a-1)x\end{array}\right|=0$

As $c_3,c_2$ are identical.