GATE DA Eigenvalues and Eigenvectors Previous Year Questions

GATE DA 2025 | Question 13

MCQ · 1 mark · −0.33

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 \(x \in \mathbb{R}^{n}\))?

The equation \(Bx = 0\) has no solution The equation \(Bx = 0\) has exactly two solutions The equation \(Bx = 0\) has infinitely many solutions The equation \(Bx = 0\) has a unique solution

Explanation

Answer: (C)

Let \(e = (1, 1, \dots, 1)^{T}\). Each row of \(A\) sums to 1, so \(Ae = e\); then \(A^{2}e = e\) and \(A^{3}e = e\) as well. Hence

\[Be = A^{3}e - 2A^{2}e + Ae = e - 2e + e = 0.\]

(Equivalently, \(B = A(A - I)^{2}\) and \((A - I)e = 0\).) So \(Bx = 0\) has the non-zero solution \(e\), and every multiple \(ce\) solves it too: infinitely many solutions.

A fails: \(x = 0\) always solves \(Bx = 0\).
B is impossible: the solutions form a subspace, so there is exactly one solution or infinitely many.
D fails: \(e \ne 0\) is a second solution.

Answer: C

GATE DA 2025 | Question 28

MSQ · 1 mark

Let \(A = I_{n} + xx^{\top}\), where \(I_{n}\) is the \(n \times n\) identity matrix and \(x \in \mathbb{R}^{n}\), \(x^{\top}x = 1\). Which of the following options is/are correct?

Rank of \(A\) is \(n\) \(A\) is invertible \(0\) is an eigenvalue of \(A\) \(A^{-1}\) has a negative eigenvalue

Explanation

Answer: (A), (B)

Find the eigenvalues of \(A\) using \(x^{\top}x = 1\):

\(Ax = x + x(x^{\top}x) = 2x\), so 2 is an eigenvalue.
Any \(v\) orthogonal to \(x\) has \(x^{\top}v = 0\), so \(Av = v\). These vectors fill an \((n - 1)\)-dimensional subspace, so 1 is an eigenvalue of multiplicity \(n - 1\).

No eigenvalue is 0, so \(A\) is invertible with rank \(n\): A and B are true and C is false. The eigenvalues of \(A^{-1}\) are the reciprocals \(\frac{1}{2}\) and 1, all positive, so D is false.

Answer: A, B

GATE DA 2024 | Question 13

MCQ · 1 mark · −0.33

Consider the matrix \(\boldsymbol{M} = \begin{bmatrix} 2 & -1 \\ 3 & 1 \end{bmatrix}\).

Which ONE of the following statements is TRUE?

The eigenvalues of \(\boldsymbol{M}\) are non-negative and real. The eigenvalues of \(\boldsymbol{M}\) are complex conjugate pairs. One eigenvalue of \(\boldsymbol{M}\) is positive and real, and another eigenvalue of \(\boldsymbol{M}\) is zero. One eigenvalue of \(\boldsymbol{M}\) is non-negative and real, and another eigenvalue of \(\boldsymbol{M}\) is negative and real.

Explanation

Answer: (B)

For a \(2 \times 2\) matrix the characteristic equation is \(\lambda^{2} - (\operatorname{tr}\boldsymbol{M})\lambda + \det \boldsymbol{M} = 0\). Here \(\operatorname{tr}\boldsymbol{M} = 2 + 1 = 3\) and \(\det \boldsymbol{M} = 2 \cdot 1 - (-1) \cdot 3 = 5\), so

\[\lambda^{2} - 3\lambda + 5 = 0, \qquad \lambda = \frac{3 \pm \sqrt{9 - 20}}{2} = \frac{3 \pm i\sqrt{11}}{2}.\]

The discriminant is negative, so the eigenvalues form a complex conjugate pair and none is real, which rules out (A), (C) and (D). A zero eigenvalue would also need \(\det \boldsymbol{M} = 0\).

Answer: B