GATE CS 2025 Set 1 Q31 — Eigenvalues and Eigenvectors, LU Decomposition

To find the eigenvalues of the matrix $A^{13}$, we first determine the eigenvalues of the matrix $A$.

The given matrix is: $$A = \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}$$

The characteristic equation of $A$ is given by $\det(A - \lambda I) = 0$: $$\det\begin{bmatrix} 1 - \lambda & 1 \\ 1 & -1 - \lambda \end{bmatrix} = 0$$ $$(1 - \lambda)(-1 - \lambda) - 1 = 0$$ $$\lambda^2 - 1 - 1 = 0 \implies \lambda^2 = 2$$

Thus, the eigenvalues of $A$ are: $$\lambda_1 = \sqrt{2} \quad \text{and} \quad \lambda_2 = -\sqrt{2}$$

Using the spectral mapping theorem, if a matrix $A$ has eigenvalues $\lambda$, then the matrix $A^k$ has eigenvalues $\lambda^k$. For $k = 13$, the eigenvalues of $A^{13}$ are:

1. $$\lambda_1^{13} = (\sqrt{2})^{13} = (2^{1/2})^{13} = 2^{6.5} = 2^6 \cdot 2^{1/2} = 64\sqrt{2}$$
2. $$\lambda_2^{13} = (-\sqrt{2})^{13} = -(\sqrt{2})^{13} = -64\sqrt{2}$$

Therefore, the eigenvalues of $A^{13}$ are $64\sqrt{2}$ and $-64\sqrt{2}$.