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23 votes
23 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. $1$
  2. $2$
  3. $n$
  4. Depends on the value of $a$
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Best answer
28 votes
28 votes

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

$R_2 \rightarrow R_2-R_1 , R_3 \rightarrow R_3-R_1 , R_4 \rightarrow R_4-R_1,$ 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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19 votes
19 votes
Ans is A.

we can eliminate all other rows using row 1. in the last only 1 row will be left.

rank = no of non zero rows = 1
Answer:

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