GATE Electrical 2026 | Question: 46

Question

Consider the system of linear equations: $A \boldsymbol{x}=\boldsymbol{b}$, where $A$ is an $n \times n$ matrix, and $\boldsymbol{x}$ and $\boldsymbol{b}$ are $n$-dimensional column vectors.

Suppose this system of equations has a unique solution. Which of the following statements is/are correct?

• A. $A^{-1}$ exists
• B. The system of equations $A^{m} \boldsymbol{x}=\boldsymbol{b}$ also has a unique solution for $m=1,2,3, \ldots$
• C. $\operatorname{rank}(A)=\operatorname{rank}\left(A^{m}\right)$, for $m=1,2,3, \ldots$
• D. $\operatorname{rank}(A)<\operatorname{rank}([A \mid \boldsymbol{b}])$, where $[A \mid \boldsymbol{b}]$ denotes the augmented matrix.

Answer 1

For a square system Ax=b to have a unique solution:

rank(A)=n​

Therefore A is invertible:

A−1 exists​

So A is correct. 

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B: Amx=b has a unique solution

Since A is invertible, any positive power is also invertible:

Am=A⋅A⋯A

and

(Am)−1=(A−1)m

Therefore Am is invertible, so:

Amx=b

has a unique solution for every m=1,2,3,…

B is correct

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C: rank(A)=rank(Am)

Since A is invertible:

rank(A)=n

And Am is also invertible:

rank(Am)=n

Therefore:

rank(A)=rank(Am)​

C is correct

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D: rank(A)<rank([A∣b])

This means the augmented matrix has a higher rank:

rank(A)<rank([A∣b])

That would mean the system is inconsistent (no solution).

But we are given the system has a unique solution.

Therefore:

rank(A)=rank([A∣b])=n

D is false