Let A = [ 2 - 1 1 1 ] . If the sum of the diagonal elements of A 13 is 3 n , then n is equal to ________ . [2024]

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Solution

Q.
Let $A = [\begin{matrix} 2 & - 1 \\ 1 & 1 \end{matrix}] .$ If the sum of the diagonal elements of $A^{13}$ is $3^{n}$ , then $n$ is equal to ________ . [2024]

Ans.
(7)

We have, $A = [\begin{matrix} 2 & - 1 \\ 1 & 1 \end{matrix}]$

$A^{2} = [\begin{matrix} 2 & - 1 \\ 1 & 1 \end{matrix}] [\begin{matrix} 2 & - 1 \\ 1 & 1 \end{matrix}] = [\begin{matrix} 3 & - 3 \\ 3 & 0 \end{matrix}]$

$A^{3} = [\begin{matrix} 3 & - 3 \\ 3 & 0 \end{matrix}] [\begin{matrix} 2 & - 1 \\ 1 & 1 \end{matrix}] = [\begin{matrix} 3 & - 6 \\ 6 & - 3 \end{matrix}]$

$A^{4} = [\begin{matrix} 3 & - 3 \\ 3 & 0 \end{matrix}] [\begin{matrix} 3 & - 3 \\ 3 & 0 \end{matrix}] = [\begin{matrix} 0 & - 9 \\ 9 & - 9 \end{matrix}]$

$A^{6} = [\begin{matrix} 3 & - 3 \\ 3 & 0 \end{matrix}] [\begin{matrix} 0 & - 9 \\ 9 & - 9 \end{matrix}] = [\begin{matrix} - 27 & 0 \\ 0 & - 27 \end{matrix}]$

$A^{6} = - 27 I$

$A^{12} = A^{6} A^{6} = (- 27 I) \times (- 27) I$

$\Rightarrow A^{12} = [\begin{matrix} 729 & 0 \\ 0 & 729 \end{matrix}] = 729 I$

So, $A^{13} = 729 I \cdot A = 3^{6} \cdot A = 3^{6} [\begin{matrix} 2 & - 1 \\ 1 & 1 \end{matrix}] = [\begin{matrix} 3^{6} \times 2 & 3^{6} (- 1) \\ 3^{6} & 3^{6} \end{matrix}]$

Sum of diagonal elements $= 3^{6} \times 2 + 3^{6} = 3^{6} \times 3 = 3^{7}$

$\Rightarrow n = 7$

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