GATE CS 2025 Set 2 Q1 — Linear Algebra: Matrices, Determinants, System of Linear Equations

To find $A^8$, we first compute $A^2$ by multiplying the matrix $A$ with itself:

$$A^2 = \begin{pmatrix} 1 & 2 \\ 2 & -1 \end{pmatrix} \begin{pmatrix} 1 & 2 \\ 2 & -1 \end{pmatrix}$$

Performing the matrix multiplication:

$$A^2 = \begin{pmatrix} (1)(1) + (2)(2) & (1)(2) + (2)(-1) \\ (2)(1) + (-1)(2) & (2)(2) + (-1)(-1) \end{pmatrix} = \begin{pmatrix} 5 & 0 \\ 0 & 5 \end{pmatrix} = 5I$$

where $I$ is the $2 \times 2$ identity matrix.

Now, we can compute $A^8$ by raising $A^2$ to the power of $4$:

$$A^8 = (A^2)^4 = (5I)^4 = 5^4 I^4 = 625I$$

Thus,

$$A^8 = \begin{pmatrix} 625 & 0 \\ 0 & 625 \end{pmatrix}$$

Therefore, the correct option is C.