GATE2025_CS1_Q41

Let A be a 2 × 2 matrix as given.

A = [ 1   1
1   -1 ]

What are the eigenvalues of the matrix A¹³ ?

✅ Final Answer:

The correct answer is option (D) 64√2, -64√2.

🔹 Short Answer:

We use the property that if λ is an eigenvalue of a matrix A, then λᵏ is an eigenvalue of the matrix Aᵏ. First, we find the eigenvalues of A by solving the characteristic equation `det(A - λI) = 0`, which gives λ² - 2 = 0, so λ = ±√2. Then, the eigenvalues of A¹³ are (√2)¹³ = 64√2 and (-√2)¹³ = -64√2.

🔸 Long Answer:

Calculating the matrix A¹³ directly and then finding its eigenvalues would be extremely time-consuming. Instead, we use a fundamental property of eigenvalues.

Step 1: State the Key Eigenvalue Property

For any square matrix A, if λ is an eigenvalue of A, then λᵏ is an eigenvalue of the matrix Aᵏ, where k is a positive integer. Our strategy is to find the eigenvalues of A first and then raise them to the power of 13.

Step 2: Find the Eigenvalues of Matrix A

The eigenvalues (λ) are the roots of the characteristic equation, which is given by det(A - λI) = 0, where I is the identity matrix.

A - λI = [ 1   1
1   -1 ] - λ [ 1   0
0   1 ] = [ 1-λ   1
1   -1-λ ]

Now, we compute the determinant:

det(A - λI) = (1 - λ)(-1 - λ) - (1)(1)
= -(1 - λ)(1 + λ) - 1
= -(1 - λ²) - 1
= -1 + λ² - 1
= λ² - 2

Set the characteristic equation to zero to find the eigenvalues:

λ² - 2 = 0  ⟹  λ² = 2  ⟹  λ = ±√2

So, the eigenvalues of matrix A are λ₁ = √2 and λ₂ = -√2.

Step 3: Calculate the Eigenvalues of A¹³

Using the property from Step 1, the eigenvalues of A¹³ are (λ₁)¹³ and (λ₂)¹³.

• First eigenvalue: (√2)¹³ = (2^1/2)¹³ = 2^13/2 = 2^6.5 = 2⁶ × 2^1/2 = 64√2
• Second eigenvalue: (-√2)¹³ = (-1)¹³ × (√2)¹³ = -1 × (64√2) = -64√2

Conclusion

The eigenvalues of A¹³ are 64√2 and -64√2, which corresponds to option (D).

Bibliography:
- Strang, G. (2016). Introduction to Linear Algebra. Wellesley-Cambridge Press. (Chapter 6: Eigenvalues and Eigenvectors).
- Kreyszig, E. (2011). Advanced Engineering Mathematics. Wiley. (Chapter 8: Linear Algebra: Matrix Eigenvalue Problems).

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