Question

If \( A = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \), \( P = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} \) and \( X = A P A^T \), then \( A^T X^{50} A \) is:

• A. \( \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \)
• B. \( \begin{bmatrix} 2 & 1 \\ 0 & -1 \end{bmatrix} \)
• C. \( \begin{bmatrix} 25 & 1 \\ 1 & -25 \end{bmatrix} \)
• D. \( \begin{bmatrix} 1 & 50 \\ 0 & 1 \end{bmatrix} \)

Hint:

For a matrix of the form \( P = \begin{bmatrix} 1 & k \\ 0 & 1 \end{bmatrix} \), its \( n \)th power is given by: \[ P^n = \begin{bmatrix} 1 & nk \\ 0 & 1 \end{bmatrix}. \]