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

\[ A = \begin{bmatrix} 2 & -1 \\ 1 & 1 \end{bmatrix} \] \[ A^2 = \begin{bmatrix} 2 & -1 \\ 1 & 1 \end{bmatrix} \times \begin{bmatrix} 2 & -1 \\ 1 & 1 \end{bmatrix} \] \[ A^2 = \begin{bmatrix} 3 & -3 \\ 3 & 0 \end{bmatrix} \] \[ A^3 = \begin{bmatrix} 3 & -6 \\ 6 & -3 \end{bmatrix} \] \[ A^4 = \begin{bmatrix} 0 & -9 \\ -9 & -9 \end{bmatrix} \] \[ A^5 = \begin{bmatrix} -9 & -9 \\ 9 & -18 \end{bmatrix} \] \[ A^8 = \begin{bmatrix} 0 & -9 \\ 9 & -9 \end{bmatrix} \] \[ A^{13} = A^8 \times A^5 = \begin{bmatrix} 81 & 81 \\ -81 & 0 \end{bmatrix} \times \begin{bmatrix} -9 & -9 \\ 9 & -18 \end{bmatrix} \] \[ A^{13} = \left[ (-81)(-9) + (81 \times 9) \right] \quad \left[ (-81)(9) \right] \] \[ \text{Sum of diagonal} = (81 \times 27) = 34^3 \times 3^7 \] \[ \Rightarrow n = 7 \] 

Answer 2

The given matrix is:
\[ A = \begin{bmatrix} 2 & -1 \\ 1 & 1 \end{bmatrix}. \]
Compute successive powers of \(A\):
\[ A^2 = \begin{bmatrix} 2 & -1 \\ 1 & 1 \end{bmatrix} \begin{bmatrix} 2 & -1 \\ 1 & 1 \end{bmatrix} = \begin{bmatrix} 3 & -3 \\ 3 & 0 \end{bmatrix}. \]
\[ A^3 = A^2 \cdot A = \begin{bmatrix} 3 & -3 \\ 3 & 0 \end{bmatrix} \begin{bmatrix} 2 & -1 \\ 1 & 1 \end{bmatrix} = \begin{bmatrix} 3 & -6 \\ 6 & -3 \end{bmatrix}. \]
\[ A^4 = A^3 \cdot A = \begin{bmatrix} 3 & -6 \\ 6 & -3 \end{bmatrix} \begin{bmatrix} 2 & -1 \\ 1 & 1 \end{bmatrix} = \begin{bmatrix} 0 & -9 \\ 9 & -9 \end{bmatrix}. \]
\[ A^5 = A^4 \cdot A = \begin{bmatrix} 0 & -9 \\ 9 & -9 \end{bmatrix} \begin{bmatrix} 2 & -1 \\ 1 & 1 \end{bmatrix} = \begin{bmatrix} -9 & -9 \\ 9 & -18 \end{bmatrix}. \]
\[ A^6 = A^5 \cdot A = \begin{bmatrix} -9 & -9 \\ 9 & -18 \end{bmatrix} \begin{bmatrix} 2 & -1 \\ 1 & 1 \end{bmatrix} = \begin{bmatrix} -27 & 0 \\ 0 & -27 \end{bmatrix}. \]
\[ A^7 = A^6 \cdot A = \begin{bmatrix} -27 & 0 \\ 0 & -27 \end{bmatrix} \begin{bmatrix} 2 & -1 \\ 1 & 1 \end{bmatrix} = \begin{bmatrix} 36 & -27 \\ -27 & 36 \end{bmatrix}. \]
Observe that the diagonal elements of \(A^7\) are \(36\) and \(36\). Their sum is:
\[ \text{Sum of diagonal elements} = 36 + 36 = 72 = 3^2 \cdot 3^5 = 3^7. \]
Thus:\[ n = 7. \]