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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 $a$
asked in Linear Algebra by Veteran (59.6k points) | 755 views

4 Answers

+15 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.

answered by Boss (40.7k points)
edited by
0
only 1 linearly independent row hence rank=1
+16 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
answered by Loyal (8.3k points)
+6 votes
All the rows of the given matrix are same. So the matrix has only one independent row.
answered by (357 points)
+4 votes
Rank of a matrix = No. of independent row (or columns) of the matrix.

i.e. Ans- A
answered by (259 points)

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