GATE 2025 DA Q.13: solution and answer

GATE 2025 · DAQ.13 · 1 mark · MCQ

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

(A)
The equation $B x = 0$ has no solution
(B)
The equation $B x = 0$ has exactly two solutions
(C)
The equation $B x = 0$ has infinitely many solutions
(D)
The equation $B x = 0$ has a unique solution

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Answer and solution

ANSWER

Official answer: CSource: Official answer key, GATE 2025 DA (IIT Roorkee)

Let $1$ be the all-ones vector in $R^{n}$ . Every row of $A$ sums to 1, so the $i$ -th entry of $A 1$ is $\sum_{j} a_{ij} = 1$ . Hence $A 1 = 1$ , and $1$ is an eigenvalue of $A$ with eigenvector $1 \neq = 0$ .
If $A v = λ v$ , then $A^{k} v = λ^{k} v$ for every $k \geq 1$ , so $p (A) v = p (λ) v$ for any polynomial $p$ . Here $p (λ) = λ^{3} - 2 λ^{2} + λ = λ (λ - 1)^{2}$ .
At $λ = 1$ : $p (1) = 1 \cdot 0 = 0$ . Therefore $B 1 = p (1) 1 = 0$ with $1 \neq = 0$ .
So $B$ has a nonzero vector in its null space. Hence $det B = 0$ , $B$ is singular, and $dim ker B \geq 1$ .
$x = 0$ always satisfies $B x = 0$ , so (A) is false. The null space is nontrivial, so $x = 0$ is not the only solution and (D) is false.
The solutions of $B x = 0$ form a subspace of $R^{n}$ . Every scalar multiple $c 1$ with $c \in R$ is a solution, and these are all different, so there are infinitely many solutions. The number of solutions can never be exactly two, so (B) is false. The answer is (C).

KEY IDEA

A matrix whose rows each sum to 1 has eigenvalue 1 with eigenvector $1$ . The polynomial $λ^{3} - 2 λ^{2} + λ = λ (λ - 1)^{2}$ vanishes at $λ = 1$ , so $B$ has a zero eigenvalue. A square matrix with a zero eigenvalue is singular, and a homogeneous system with a singular matrix has infinitely many solutions.

WATCH OUT FOR

Assuming $B x = 0$ must have only the trivial solution, or that it has no solution. A homogeneous system always has $x = 0$ as a solution. Also, a linear system over the reals never has exactly two solutions: it has 0, 1 or infinitely many.

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GATE 2025 DA Q.13: solution and answer

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