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Eigenvalues, Eigenvectors and Matrix Powers PYQs for GATE DA

Solve 3+ Eigenvalues, Eigenvectors and Matrix Powers previous year questions for GATE DA with answers and detailed solutions. Free sample questions below.

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Question 1

2026 PYQ

Level 3: Exam Standard

Let M = (cos θ s in θ - s in θ cos θ) be a 2 \times 2 matrix, where θ = \frac{2 π}{5}, and I_{2} = (1001) . Which of the following options is equal to M^{2026} ?

Question 2

2026 PYQ

Level 3: Exam Standard

Let γ_{1}, γ_{2}, γ_{3} be the eigenvalues of the matrix 100 0 cos t - s in t 0 s in t cos t, where t \in [- π, π] is in radians. Which one of the following options lists all the possible values of t satisfying γ_{1} + γ_{2} + γ_{3} = 1 + 2 ?

Question 3

2024 PYQ

Level 3: Exam Standard

Consider the matrix M = [23 - 1 1] . Which ONE of the following statements is TRUE?

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Eigenvalues, Eigenvectors and Matrix Powers PYQs for GATE DA

Solve 3+ Eigenvalues, Eigenvectors and Matrix Powers previous year questions for GATE DA with answers and detailed solutions. Free sample questions below.

Chapter Roadmap: Eigenvalues, Eigenvectors and Matrix Powers

Chapter Roadmap

Eigenvalues, Eigenvectors and Matrix Powers

Rotation Matrices & Matrix Powers

Geometric intuition, 2D/3D rotations, computing $A^{n}$ without diagonalization, and trace-eigenvalue relations. Very High Weightage.

Characteristic Polynomial & Nature of Eigenvalues

Building $det (A - λ I) = 0$ , real vs complex roots, and Cayley-Hamilton theorem. Moderate Weightage.

By the end: Instantly compute $A^{2026}$ , find sums of eigenvalue powers via trace, and predict eigenvalue nature from matrix structure.

The Geometry of Matrices: Rotations and Powers

The Geometry of Matrices

A matrix is a transformation of space. The purest rigid transformation is a rotation.

The Core Intuition

If a matrix $A$ rotates a vector by $θ$ , then $A^{2}$ rotates it by $2 θ$ , and $A^{n}$ rotates it by $n θ$ .

Why this matters for exams

When you see a rotation matrix, do not diagonalize it to find $A^{n}$ . Diagonalization involves complex numbers and messy algebra. Just multiply the angle by $n$ .

This turns a brutal $O (n)$ matrix multiplication into a simple trigonometric evaluation.

Eigenvalues, Eigenvectors and Matrix Powers: Solved Questions with Step-by-Step Explanations (3 Problems)

Question 1 · Linear Algebra · 2026 MCQ

Let

M = (cos θ s in θ - s in θ cos θ)

be a

2 \times 2

matrix, where

θ = \frac{2 π}{5}

, and

I_{2} = (1001)

.
Which of the following options is equal to

M^{2026}

A.
$M^{2}$
B.
$M$
C.
$M^{- 1}$
D.
$I_{2}$

Correct Answer:

Step-by-Step Solution

Insight: $M = R (2 π /5)$ is a rotation by $2 π /5$ , so $M^{5} = I$ . Reduce the exponent $2026$ modulo $5$ .

Exam route: $2026 = 5 \times 405 + 1$ , so $M^{2026} = (M^{5})^{405} \cdot M^{1} = I^{405} \cdot M = M$ .

Learning route:

Identify $M$ as the standard counter-clockwise rotation matrix $R (θ)$ with $θ = \frac{2 π}{5}$ .
A rotation by $θ$ raised to the $n$ -th power is a rotation by $n θ$ : $M^{n} = R (n θ)$ .
Compute $n θ = 2026 \times \frac{2 π}{5} = \frac{4052 π}{5}$ .
Reduce modulo $2 π$ : $\frac{4052}{5} = 810 + \frac{2}{5}$ , so $\frac{4052 π}{5} = 810 π + \frac{2 π}{5} = 405 (2 π) + \frac{2 π}{5}$ .
The effective angle is $\frac{2 π}{5} = θ$ , so $M^{2026} = R (θ) = M$ .
Alternatively, $M^{5} = R (5 \cdot 2 π /5) = R (2 π) = I$ . Then $2026 mod 5 = 1$ , so $M^{2026} = M^{1} = M$ .

Question 2 · Linear Algebra · 2026 MCQ

Let

γ_{1}, γ_{2}, γ_{3}

be the eigenvalues of the matrix

100 0 cos t - s in t 0 s in t cos t

, where

t \in [- π, π]

is in radians.

Which one of the following options lists all the possible values of

t

satisfying

γ_{1} + γ_{2} + γ_{3} = 1 + 2

A.
${\frac{π}{3}, - \frac{π}{4}}$
B.
${\frac{π}{4}, - \frac{π}{3}}$
C.
${\frac{π}{4}, - \frac{π}{4}}$
D.
${\frac{π}{3}, - \frac{π}{3}}$

Correct Answer:

Step-by-Step Solution

Insight: The matrix is block-diagonal — a $1 \times 1$ block $[1]$ and a $2 \times 2$ rotation block — so the eigenvalues are the union of the blocks' eigenvalues.

Exam route: Eigenvalues of the $1 \times 1$ block: ${1}$ . Eigenvalues of the $2 \times 2$ block $[cos t - sin t sin t cos t]$ : solve $(cos t - λ)^{2} + sin^{2} t = 0 \Rightarrow λ = cos t \pm i sin t = e^{\pm i t}$ . Sum of all three eigenvalues (the trace) is $1 + 2 cos t$ . Set $1 + 2 cos t = 1 + 2 \Rightarrow cos t = \frac{1}{2}$ . For $t \in [- π, π]$ , this gives $t = \pm \frac{π}{4}$ .

Learning route:

Identify the block structure: $A = diag (1, R (- t))$ where $R (- t)$ is a clockwise rotation by $t$ .
The spectrum of a block-diagonal matrix is the union of the spectra of its blocks.
Block 1 contributes $γ_{1} = 1$ .
Block 2 has characteristic polynomial $λ^{2} - 2 cos t \cdot λ + 1 = 0$ , giving $γ_{2} = e^{i t}, γ_{3} = e^{- i t}$ .
Sum: $γ_{1} + γ_{2} + γ_{3} = 1 + e^{i t} + e^{- i t} = 1 + 2 cos t$ .
Equation: $1 + 2 cos t = 1 + 2 \Rightarrow cos t = \frac{2}{2}$ .
In $[- π, π]$ , $cos t = \frac{2}{2}$ has exactly two solutions: $t = \frac{π}{4}$ and $t = - \frac{π}{4}$ .
The set of all such $t$ is ${\frac{π}{4}, - \frac{π}{4}}$ , matching option C.

Question 3 · Linear Algebra · 2024 MCQ

Consider the matrix

M = [23 - 1 1]

.
Which ONE of the following statements is TRUE?

A.
The eigenvalues of $M$ are non-negative and real.
B.
The eigenvalues of $M$ are complex conjugate pairs.
C.
One eigenvalue of $M$ is positive and real, and another eigenvalue of $M$ is zero.
D.
One eigenvalue of $M$ is non-negative and real, and another eigenvalue of $M$ is negative and real.

Correct Answer:

Step-by-Step Solution

Insight: For a $2 \times 2$ matrix, the characteristic equation is $λ^{2} - Tr (M) λ + det (M) = 0$ . The discriminant determines the nature of the eigenvalues.

Exam route: $Tr (M) = 2 + 1 = 3$ , $det (M) = 2 (1) - (- 1) (3) = 5$ . Equation: $λ^{2} - 3 λ + 5 = 0$ . Discriminant $Δ = 9 - 20 = - 11 < 0$ . So eigenvalues are a complex conjugate pair.

Learning route:

Compute trace: $Tr (M) = 2 + 1 = 3$ .
Compute determinant: $det (M) = (2) (1) - (- 1) (3) = 2 + 3 = 5$ .
Characteristic equation: $λ^{2} - 3 λ + 5 = 0$ .
Discriminant: $Δ = (- 3)^{2} - 4 (1) (5) = 9 - 20 = - 11$ .
Since $Δ < 0$ , the roots are complex conjugates: $λ = \frac{3 \pm i 11}{2}$ .
This matches option B: "The eigenvalues of $M$ are complex conjugate pairs."
Note: $M$ is a real matrix but not symmetric, so there is no guarantee of real eigenvalues. This is the key insight — real entries do not imply real eigenvalues.

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Notes written for recall, not for volume

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Honest progress

Unlock the whole course

Eigenvalues, Eigenvectors and Matrix Powers PYQs for GATE DA

Chapter Roadmap: Eigenvalues, Eigenvectors and Matrix Powers

Chapter Roadmap

Rotation Matrices & Matrix Powers

Characteristic Polynomial & Nature of Eigenvalues

The Geometry of Matrices: Rotations and Powers

The Geometry of Matrices

The Core Intuition

Why this matters for exams

Eigenvalues, Eigenvectors and Matrix Powers: Solved Questions with Step-by-Step Explanations (3 Problems)

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

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