Eigenvalues, Eigenvectors and Matrix Powers Notes for GATE DA

Eigenvalues, Eigenvectors and Matrix Powers notes for GATE DA: 17 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

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.

The 2D Rotation Matrix $R( heta)$

The 2D Rotation Matrix $R (θ)$

A counter-clockwise rotation by $θ$ in the 2D plane:

R (θ) = (cos θ sin θ - sin θ cos θ)

Orthogonal

R (θ)^{T} R (θ) = I

. Inverse is

R (- θ)

Determinant

det (R (θ)) = cos^{2} θ + sin^{2} θ = 1

Trace

Tr (R (θ)) = 2 cos θ

Identification: $2 \times 2$ matrix with equal diagonal elements, and off-diagonal elements equal in magnitude but opposite in sign.

Computing Powers of $R( heta)$

Computing Powers of $R (θ)$

[R (θ)]^{n} = R (n θ) = (cos (n θ) sin (n θ) - sin (n θ) cos (n θ))

The Exam Method for Large $n$

Multiply: Calculate $α = n θ$ .
Reduce: Find remainder of $α$ divided by $2 π$ . Let this be $α_{reduced}$ .
Evaluate: Plug $α_{reduced}$ into the standard format.

Example: $M^{2026}$ where $M = R (2 π /5)$

$α = 2026 \times \frac{2 π}{5} = 810.4 π$ .
Reduce modulo $2 π$ : $810.4 π = 405 (2 π) + 0.4 π$ .
Effective angle is $0.4 π = \frac{2 π}{5}$ . Thus, $M^{2026} = M$ .

Eigenvalues, Eigenvectors and Matrix Powers: Solved Questions with Step-by-Step Explanations (10 Problems)

Question 1 · Linear Algebra MCQ

Let $R (θ)$ be a 2D rotation matrix with $θ = π /4$ . What is the minimum positive integer $n$ such that $[R (θ)]^{n} = I$ ?

A.
4
B.
6
C.
8
D.
16

Correct Answer:

Step-by-Step Solution

Key idea: This is a contradiction-style question where you must find the smallest $n$ satisfying a condition.

Step 1: Recall $[R (θ)]^{n} = R (n θ)$ .

Step 2: For $R (n θ) = I$ , we need $n θ = 2 k π$ for some positive integer $k$ .

Step 3: Substitute $θ = π /4$ : $n \cdot \frac{π}{4} = 2 k π$ .

Step 4: Simplify: $n = 8 k$ .

Step 5: The minimum positive integer $n$ occurs at $k = 1$ , giving $n = 8$ .

Answer: C

Common trap: Students might pick $n = 4$ because $4 \times π /4 = π$ , but $R (π) = - I \neq = I$ .

Question 2 · Linear Algebra MCQ

What is the maximum possible value of the trace of a $2 \times 2$ rotation matrix $R (θ)$ ?

A.
0
B.
1
C.
2
D.
It is unbounded

Correct Answer:

Step-by-Step Solution

Key idea: This is an observation question on the trace formula for rotation matrices.

Step 1: Recall the trace of a 2D rotation matrix: $Tr (R (θ)) = 2 cos θ$ .

Step 2: The cosine function satisfies $- 1 \leq cos θ \leq 1$ for all real $θ$ .

Step 3: Therefore, $2 cos θ$ satisfies $- 2 \leq 2 cos θ \leq 2$ .

Step 4: The maximum value is $2$ , achieved when $cos θ = 1$ , i.e., $θ = 0$ .

Answer: C

Common trap: Students might think the trace can be larger if they forget the bound on cosine, or confuse it with the determinant (which is always 1).

Question 3 · Linear Algebra MCQ

Below are two statements about a $3 \times 3$ matrix representing rotation about the $x$ -axis by angle $t$ :

\textbf{Assertion (A):} The trace of this matrix is $1 + 2 cos (t)$ .

\textbf{Reason (R):} The eigenvalues of this matrix are $1$ , $cos (t)$ , and $sin (t)$ .

Which option is correct?

A.
Both A and R are true, and R is the correct explanation of A.
B.
Both A and R are true, but R is not the correct explanation of A.
C.
A is true but R is false.
D.
A is false but R is true.

Correct Answer:

Step-by-Step Solution

Key idea: This is a construction-style assertion-reason question on 3D rotation matrices.

Step 1: Recall the standard form of a 3D rotation about the $x$ -axis:

$A = 100 0 cos t sin t 0 - sin t cos t$

Step 2: Evaluate Assertion (A):

$Tr (A) = 1 + cos t + cos t = 1 + 2 cos t$ . So A is TRUE.

Step 3: Evaluate Reason (R):

The matrix is block diagonal: block 1 is $[1]$ , block 2 is $R (t)$ .

Eigenvalues of block 1: $λ_{1} = 1$ .

Eigenvalues of block 2: $λ_{2} = e^{i t}$ , $λ_{3} = e^{- i t}$ .

So eigenvalues are $1, e^{i t}, e^{- i t}$ , NOT $1, cos t, sin t$ .

Therefore R is FALSE.

Answer: C

Common trap: Students might confuse matrix entries ( $cos t, sin t$ ) with eigenvalues, leading them to accept R as true.

Question 4 · Linear Algebra MCQ

Let $M = R (\frac{2 π}{5})$ be a 2D rotation matrix. What is the minimum positive integer $n > 1$ such that $M^{n} = M$ ?

A.
5
B.
6
C.
10
D.
11

Correct Answer:

Step-by-Step Solution

Key idea: This is a contradiction-style question where you must find when $M^{n}$ returns to $M$ .

Step 1: Recall $M^{n} = R (n \cdot \frac{2 π}{5})$ .

Step 2: For $M^{n} = M$ , we need $R (n \cdot \frac{2 π}{5}) = R (\frac{2 π}{5})$ .

Step 3: Two rotation matrices are equal iff their angles differ by a multiple of $2 π$ :

$n \cdot \frac{2 π}{5} = \frac{2 π}{5} + 2 k π for some integer k$

Step 4: Simplify:

$(n - 1) \cdot \frac{2 π}{5} = 2 k π$

$(n - 1) \cdot \frac{2}{5} = 2 k$

$n - 1 = 5 k$

$n = 5 k + 1$

Step 5: For $n > 1$ , the minimum occurs at $k = 1$ , giving $n = 6$ .

Step 6: Verify: $M^{6} = R (6 \cdot \frac{2 π}{5}) = R (\frac{12 π}{5}) = R (2 π + \frac{2 π}{5}) = R (\frac{2 π}{5}) = M$ . ✓

Answer: B

Common trap: Students might pick $n = 5$ because $M^{5} = R (2 π) = I \neq = M$ , or they might think $n = 1$ works (but the question specifies $n > 1$ ).

Question 5 · Linear Algebra MCQ

Below are two statements about a $3 \times 3$ matrix $A = 100 0 cos t sin t 0 - sin t cos t$ representing rotation about the $x$ -axis by angle $t$ :

\textbf{Assertion (A):} The trace of $A$ is $1 + 2 cos (t)$ .

\textbf{Reason (R):} The determinant of $A$ is $1$ .

Which option is correct?

A.
Both A and R are true, and R is the correct explanation of A.
B.
Both A and R are true, but R is not the correct explanation of A.
C.
A is true but R is false.
D.
A is false but R is true.

Correct Answer:

Step-by-Step Solution

Key idea: This is a construction-style assertion-reason question testing understanding of trace vs determinant for 3D rotation matrices.

Step 1: Evaluate Assertion (A):

The trace is the sum of diagonal elements:

$Tr (A) = 1 + cos t + cos t = 1 + 2 cos t$

So A is TRUE.

Step 2: Evaluate Reason (R):

The determinant of a block diagonal matrix is the product of determinants of blocks:

$det (A) = det ([1]) \cdot det (cos t sin t - sin t cos t) = 1 \cdot (cos^{2} t + sin^{2} t) = 1$

So R is TRUE.

Step 3: Does R explain A?

The trace and determinant are independent properties of a matrix. Knowing that $det (A) = 1$ does not tell us what $Tr (A)$ is. For example, a matrix could have determinant $1$ but trace $0$ , $3$ , or any other value.

Therefore, R does NOT explain A.

Answer: B

Common trap: Students might think that since both statements are true and both relate to the matrix, R must explain A. But trace and determinant are separate invariants.

Question 6 · Linear Algebra MCQ

Let $A$ be a $2 \times 2$ matrix with eigenvalues $1$ and $- 2$ . What is the value of $Tr (A^{3})$ ?

A.
-7
B.
7
C.
-1
D.
9

Correct Answer:

Step-by-Step Solution

Key idea: This is a direct formula application on the trace-eigenvalue relation for matrix powers.

Step 1: Recall the fundamental bridge between trace and eigenvalues: $Tr (M) = \sum λ_{i}$ .

Step 2: When a matrix is raised to a power $n$ , its eigenvalues are also raised to the power $n$ . Thus, the eigenvalues of $A^{3}$ are $1^{3} = 1$ and $(- 2)^{3} = - 8$ .

Step 3: Apply the trace formula to $A^{3}$ :

$Tr (A^{3}) = 1 + (- 8) = - 7$

Answer: A

Common trap: Students often confuse $Tr (A^{3})$ with $(Tr (A))^{3} = (- 1)^{3} = - 1$ , or they might incorrectly calculate $1^{3} + 2^{3} = 9$ by ignoring the negative sign.

Question 7 · Linear Algebra MCQ

Let $M = R (\frac{π}{4})$ be a 2D rotation matrix. What is the minimum positive integer $n$ such that $M^{n} = M^{- 2}$ ?

A.
4
B.
6
C.
8
D.
10

Correct Answer:

Step-by-Step Solution

Key idea: This is a contradiction-style question where the condition $M^{n} = M^{- 2}$ requires careful angle matching.

Step 1: Recall that $M = R (π /4)$ , so $M^{n} = R (nπ /4)$ .

Step 2: The inverse property gives $M^{- 2} = R (- 2 π /4) = R (- π /2)$ .

Step 3: Set the matrices equal:

$R (\frac{nπ}{4}) = R (- \frac{π}{2})$

Step 4: Two rotation matrices are equal if and only if their angles differ by a multiple of $2 π$ :

$\frac{nπ}{4} = - \frac{π}{2} + 2 k π for some integer k$

Step 5: Solve for $n$ :

$\frac{nπ}{4} = \frac{- π + 4 k π}{4}$

$n = 4 k - 2$

Step 6: Find the minimum positive integer $n$ .

For $k = 0$ : $n = - 2$ (not positive)
For $k = 1$ : $n = 4 (1) - 2 = 2$ (Wait, $M^{2} = R (π /2) \neq = R (- π /2)$ . Let me re-evaluate Step 4).

Correction: $R (α) = R (β) ⟺ α = β + 2 k π$ .

$nπ /4 = - π /2 + 2 k π ⟹ n = - 2 + 8 k$ .

For $k = 1$ , $n = 6$ .

Answer: B

Common trap: Students might solve $M^{n} = I$ (giving $n = 8$ ) or $M^{n} = M^{2}$ (giving $n = 10$ ), ignoring the negative sign in the exponent.

Question 8 · Linear Algebra MCQ

Below are two statements about a $3 \times 3$ matrix $A = 100 0 cos t sin t 0 - sin t cos t$ representing rotation about the $x$ -axis by angle $t$ :

\textbf{Assertion (A):} The determinant of $A^{2}$ is $1$ .

\textbf{Reason (R):} The eigenvalues of $A^{2}$ are $1$ , $cos (2 t)$ , and $sin (2 t)$ .

Which option is correct?

A.
A is false but R is true.
B.
A is true but R is false.
C.
Both A and R are true, and R is the correct explanation of A.
D.
Both A and R are true, but R is not the correct explanation of A.

Correct Answer:

Step-by-Step Solution

Key idea: This is a construction-style assertion-reason question testing the relationship between eigenvalues and determinant for powers of 3D rotation matrices.

Step 1: Evaluate Assertion (A):

The matrix $A$ is a rotation matrix, so $det (A) = 1$ .

Using the multiplicative property of determinants:

$det (A^{2}) = det (A \cdot A) = det (A) \cdot det (A) = 1 \cdot 1 = 1$

So, A is TRUE.

Step 2: Evaluate Reason (R):

The matrix $A$ is block diagonal with blocks $[1]$ and $R (t)$ .

The eigenvalues of $A$ are $1, e^{i t}, e^{- i t}$ .

When a matrix is squared, its eigenvalues are squared.

Thus, the eigenvalues of $A^{2}$ are $1^{2}, (e^{i t})^{2}, (e^{- i t})^{2}$ , which simplifies to $1, e^{2 i t}, e^{- 2 i t}$ .

The values $1, cos (2 t), sin (2 t)$ are incorrect (they confuse matrix entries with eigenvalues).

So, R is FALSE.

Answer: B

Common trap: Students might confuse the matrix entries of the $2 \times 2$ block with the eigenvalues, thinking the eigenvalues of $A^{2}$ are $1, cos (2 t), sin (2 t)$ . This leads them to believe R is true.

Question 9 · Linear Algebra MCQ

Consider the following statements about a

2 \times 2

real orthogonal matrix

A

with

det (A) = 1

:
P: The eigenvalues of

A

are always real.
Q: The trace of

A

lies in the interval

[- 2, 2]

.
R:

A^{n}

is also orthogonal for any positive integer

n

.
Which of the following is correct?

A.
Only P is true
B.
Only Q and R are true
C.
Only P and R are true
D.
All three are true

Correct Answer:

Step-by-Step Solution

Key idea: This tests multiple properties of rotation matrices (orthogonal with det = 1).

Step 1: Analyze statement P.

A $2 \times 2$ orthogonal matrix with $det = 1$ is a rotation matrix $R (θ)$ .
Eigenvalues are $e^{\pm i θ} = cos θ \pm i sin θ$ .
These are real only when $sin θ = 0$ (i.e., $θ = 0$ or $π$ ).
For other angles (e.g., $θ = π /2$ ), eigenvalues are complex.
Therefore, P is FALSE.

Step 2: Analyze statement Q.

$Tr (R (θ)) = 2 cos θ$ .
Since $cos θ \in [- 1, 1]$ , we have $2 cos θ \in [- 2, 2]$ .
Therefore, Q is TRUE.

Step 3: Analyze statement R.

If $A$ is orthogonal, then $A^{T} A = I$ .
$(A^{n})^{T} A^{n} = (A^{T})^{n} A^{n} = (A^{T} A)^{n} = I^{n} = I$ .
Therefore, $A^{n}$ is orthogonal. R is TRUE.

Step 4: Conclusion: Only Q and R are true.

Answer: B

Question 10 · Linear Algebra MCQ

Consider the following statements about a

3 \times 3

real matrix

A

:
P: If the sum of elements in each row is 5, then 5 is an eigenvalue of

A

.
Q: If

A

is skew-symmetric, then all its eigenvalues are strictly imaginary.
R: If

det (A) = 0

, then 0 is an eigenvalue of

A

.
Which of the following options is correct?

A.
Only P and Q are true
B.
Only P and R are true
C.
Only Q and R are true
D.
All three statements are true

Correct Answer:

Step-by-Step Solution

Key idea: This tests fundamental properties of eigenvalues related to matrix structure and determinants.

Step 1: Analyze statement P.

If the sum of each row is 5, then multiplying $A$ by the column vector of all ones $1 = (1, 1, 1)^{T}$ yields $(5, 5, 5)^{T} = 51$ .
By definition, $A 1 = 51$ , so 5 is an eigenvalue with eigenvector $1$ .
Statement P is TRUE.

Step 2: Analyze statement Q.

A real skew-symmetric matrix satisfies $A = - A^{T}$ .
Its eigenvalues are purely imaginary or zero.
For a $3 \times 3$ skew-symmetric matrix, the characteristic polynomial is a cubic with real coefficients. Complex roots must come in conjugate pairs. Since the degree is odd, there must be at least one real root.
The only real number that is "purely imaginary or zero" is 0 itself. Thus, 0 is always an eigenvalue.
Since 0 is not "strictly imaginary" (it is real), statement Q is FALSE.

Step 3: Analyze statement R.

The determinant of a matrix is the product of its eigenvalues: $det (A) = λ_{1} λ_{2} λ_{3}$ .
If $det (A) = 0$ , then $λ_{1} λ_{2} λ_{3} = 0$ , which implies at least one eigenvalue must be 0.
Statement R is TRUE.

Step 4: Conclusion: Only P and R are true.

Answer: B