The GATE Grind

GATE 2025 DA – Question 13

Linear Algebra · Eigenvalues, eigenvectors and determinant · 1 mark · Multiple choice

The sum of the elements in each row of $A \in \mathbb{R}^{n \times n}$ is 1. If $B = A^3 - 2A^2 + A$, which one of the following statements is correct (for $x \in \mathbb{R}^n$)?

  1. The equation $Bx = 0$ has no solution
  2. The equation $Bx = 0$ has exactly two solutions
  3. The equation $Bx = 0$ has infinitely many solutions
  4. The equation $Bx = 0$ has a unique solution

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Correct answer: (C) The equation $Bx = 0$ has infinitely many solutions

Explanation

Row sums of 1 mean $A\mathbf{1} = \mathbf{1}$, where $\mathbf{1}$ is the vector of ones. Then $B\mathbf{1} = (A^3 - 2A^2 + A)\mathbf{1} = (1 - 2 + 1)\mathbf{1} = 0$. So the non-zero vector $\mathbf{1}$ is in the null space of $B$, and every multiple of it is a solution of $Bx = 0$. There are infinitely many solutions.