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+20 votes

The rank of the following $(n+1) \times (n+1)$ matrix, where $a$ is a real number is $$ \begin{bmatrix} 1 & a & a^2 & \dots & a^n \\ 1 & a & a^2 & \dots & a^n \\ \vdots & \vdots & \vdots & \: & \vdots \\ \vdots & \vdots & \vdots & \: & \vdots \\ 1 & a & a^2 & \dots & a^n \end{bmatrix}$$

- $1$
- $2$
- $n$
- Depends on the value of $a$

+21 votes

Best answer

$\begin{bmatrix} 1 & a & a^2 & \dots & a^n \\ 1 & a & a^2 & \dots & a^n \\ \vdots & \vdots & \vdots & \: & \vdots \\ \vdots & \vdots & \vdots & \: & \vdots \\ 1 & a & a^2 & \dots & a^n \end{bmatrix}$

$R_2 \rightarrow R_2-R_1 , R_3 \rightarrow R_3-R_1 , R_4 \rightarrow R_4-R_1,$ and so on

$\begin{bmatrix} 1 & a & a^2 & \dots & a^n \\ 0 & 0 & 0 & \dots & 0 \\ \vdots & \vdots & \vdots & \: & \vdots \\ \vdots & \vdots & \vdots & \: & \vdots \\ 0 & 0 &0 & \dots &0 \end{bmatrix}$

Rank of the Matrix $=1$

Hence, option **(A) 1** is the correct choice.

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