Question

Let $A = \begin{bmatrix} 2 & -1 \\ 1 & 1 \end{bmatrix}$. If the sum of the diagonal elements of $A^{13}$ is $3^n$, then $n$ is equal to _____.

Answer 1

Correct Answer: 7

\[ A = \begin{bmatrix} 2 & -1 \\ 1 & 1 \end{bmatrix} \] \[ A^2 = \begin{bmatrix} 2 & -1 \\ 1 & 1 \end{bmatrix} \times \begin{bmatrix} 2 & -1 \\ 1 & 1 \end{bmatrix} = \begin{bmatrix} 3 & -3 \\ 3 & 0 \end{bmatrix} \] \[ A^3 = \begin{bmatrix} 3 & -6 \\ 6 & -3 \end{bmatrix} \] \[ A^4 = \begin{bmatrix} 0 & -9 \\ -9 & -9 \end{bmatrix} \] \[ A^5 = \begin{bmatrix} -9 & -9 \\ 9 & -18 \end{bmatrix} \] \[ A^8 = \begin{bmatrix} 0 & -9 \\ 9 & -9 \end{bmatrix} \] \[ A^{13} = A^8 \times A^5 = \begin{bmatrix} 81 & 81 \\ -81 & 0 \end{bmatrix} \times \begin{bmatrix} -9 & -9 \\ 9 & -18 \end{bmatrix} \] The matrix multiplication for \( A^{13} \) results in the element \( [ (-81)(-9) + (81 \times 9) ] \) and \( [ (-81)(9) ] \). The sum of the diagonal elements is \( (81 \times 27) \), which simplifies to \( 34^3 \times 3^7 \). Therefore, \( n = 7 \).