Let A=[array|c|c| 2 & -1 1 & 1 array].

Question

Let $A=[\begin{array}{|c|c|} 2 & -1 \ 1 & 1 \end{array}]$. If the sum of the diagonal elements of ${A}^{13}$ is ${3}^{n}$, then $n$ is equal to \_\_\_\_\_\_\_\_.

Answer: 7

Accepted Answer

$\begin{gathered} A=[\begin{gathered} 2 \quad -1 \ 1 \quad 1] \ {A}^{2}=[2 \quad -1 \ 1 \quad 1][2 \quad -1 \ 1 \quad 1] \ {A}^{2}=[3 \quad -3 \ 3 \quad 0]=3[1 \quad -1 \ 1 \quad 0] \ {A}^{4}=9[0 \quad -1 \ 1 \quad -1] \ {A}^{8}=81[-1 \quad 1 \ -1 \quad 0] \ {A}^{12}=729[1 \quad 0 \ 0 \quad 1] \ {A}^{13}=729[1 \quad 0 \ 0 \quad 1][2 \quad -1 \ 1 \quad 1] \ {A}^{13}=[1458 \quad -729 \ 729 \quad 729 \end{gathered}] \end{gathered}$

$\begin{gathered} =2187={3}^{n} \ {3}^{7}={3}^{n} \ n=7 \end{gathered}$

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