The GATE Grind

GATE 2026 EE – Question 56

Engineering Mathematics · Linear Algebra: Systems of linear equations · 2 marks · Multiple select

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?

  1. $A^{-1}$ exists
  2. The system of equations $A^m\boldsymbol x=\boldsymbol b$ also has a unique solution for $m=1,2,3,\ldots$
  3. $\text{rank}(A)=\text{rank}(A^m)$, for $m=1,2,3,\ldots$
  4. $\text{rank}(A)<\text{rank}([A\,|\,\boldsymbol b])$, where $[A\,|\,\boldsymbol b]$ denotes the augmented matrix.

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Show answer and explanation

Correct answer: (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) $\text{rank}(A)=\text{rank}(A^m)$, for $m=1,2,3,\ldots$

Explanation

A unique solution of a square system means $A$ is invertible (A). Then $A^m$ is a product of invertible matrices, hence invertible, so $A^m\boldsymbol x=\boldsymbol b$ also has a unique solution (B) and $\text{rank}(A^m)=n=\text{rank}(A)$ (C). For a consistent system, $\text{rank}(A)=\text{rank}([A|\boldsymbol b])$, so D is false.