Let A=[\cc2 -1 \ 1 1\]. If the sum of the diagonal elements of A^13 is 3^n, then n is equal to ________.

A=[\cc 2 -1 \ 1 1 \] \ A^2=[\cc 2 -1 \ 1 1 \][\cc 2 -1 \ 1 1 \] \ A^2=[\cc 3 -3 \ 3 0 \]=3[\cc 1 -1 \ 1 0 \] \ A^4=9[\ll 0 -1 \ 1 -1 \] \ A^8=81[\ll -1 1 \ -1 0 \] \ A^12=729[\ll 1 0 \ 0 1 \] \ A^13=729[\ll 1 0 \ 0 1 \][\cc 2 -1 \ 1 1 \] \ A^13=[\cc 1458 -729 \ 729 729 \] \ Sum =2187=3^n \ 3^7=3^n \ n=7 \

Let Acc2 1 1 1 If the sum of the diagonal elements of A... | null (Mathematics)

\begin{aligned} & A=\left[\begin{array}{cc} 2 & -1 \\ 1 & 1 \end{array}\right] \\ & A^2=\left[\begin{array}{cc} 2 & -1 \\ 1 & 1 \end{array}\right]\left[\begin{array}{cc} 2 & -1 \\ 1 & 1 \end{array}\right] \\ & A^2=\left[\begin{array}{cc} 3 & -3 \\ 3 & 0 \end{array}\right]=3\left[\begin{array}{cc} 1 & -1 \\ 1 & 0 \end{array}\right] \\ & A^4=9\left[\begin{array}{ll} 0 & -1 \\ 1 & -1 \end{array}\right] \\ & A^8=81\left[\begin{array}{ll} -1 & 1 \\ -1 & 0 \end{array}\right] \\ & A^{12}=729\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \\ & A^{13}=729\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]\left[\begin{array}{cc} 2 & -1 \\ 1 & 1 \end{array}\right] \\ & A^{13}=\left[\begin{array}{cc} 1458 & -729 \\ 729 & 729 \end{array}\right] \end{aligned}

\begin{aligned} & \text { Sum }=2187=3^n \\ & 3^7=3^n \\ & n=7 \end{aligned}

Explanation

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