Given matrix: $ where: \) is a 2x2 upper triangular matrix. We will explore the pattern of powers of to find . The first few powers of are: - - - - Step 2: Observe the pattern of powers. From the above calculations, we observe the following pattern: - The top-left element of is always 1. - The top-right element of increases linearly by 1 each time, i.e., . - The bottom-left element is always 0. - The bottom-right element alternates between powers of , i.e., . Therefore, the general form of is: \) . From the pattern, we see that for even , . Since is even, we have: \) . In the matrix , we have and . Therefore: \) $ Conclusion: The value of is . Thus, the correct answer is .