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GATE 2024 CH – Question 26

Engineering Mathematics · Linear Algebra: Matrix algebra, systems of linear equations, eigenvalues and eigenvectors · 1 mark · Multiple select

Consider a linear homogeneous system of equations $\mathbf{A}\mathbf{x} = \mathbf{0}$, where $\mathbf{A}$ is an $n \times n$ matrix, $\mathbf{x}$ is an $n \times 1$ vector and $\mathbf{0}$ is an $n \times 1$ null vector. Let $r$ be the rank of $\mathbf{A}$. For a non-trivial solution to exist, which of the following conditions is/are satisfied?

  1. Determinant of A = 0
  2. $r = n$
  3. $r < n$
  4. Determinant of A ≠ 0

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Show answer and explanation

Correct answer: (A) Determinant of A = 0; (C) $r < n$

Explanation

A non-trivial solution exists only when $\mathbf{A}$ is singular, that is $\det\mathbf{A} = 0$, which is the same as the rank being less than $n$.