GATE CSE 1995 | Question: 1.24

Question

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

• A. $1$
• B. $2$
• C. $n$
• D. Depends on the value of $a$

Comments

if the second row starts with a,a,a²,......,a^n
then what is the rank

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@adarsh1006 Your doubt doesn’t seem complete. Assuming all other row remains same. Rank will be 2, since 2^nd row will not be linearly dependent on any other row.

Answer 1

Rank = Number of linearly Independent rows/columns

Only 1 independent row hence rank = 1

Ans : (A)

Answer 2

We can subtract the first row from any row and the whole row will become 0..in this way only the first row will be non-zero. So the rank is 1

Answer 3

.