Question
For the matrices $A = \begin{bmatrix} 3 & - 4 \\ 1 & - 1 \end{bmatrix}$ and $B = \begin{bmatrix} - 29 & 49 \\ - 13 & 18 \end{bmatrix}$, if $\left( A^{15} + B \right)\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$, then among the following which one is true?
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Solution
Here $A^{n} = \begin{bmatrix} 2n + 1 & - 4n \\ n & - 2n + 1 \end{bmatrix}$
$${\Rightarrow A^{15} = \begin{bmatrix} 31 & - 60 \\ 15 & - 29 \end{bmatrix} }{\Rightarrow A^{15} + B = \begin{bmatrix} 2 & - 11 \\ 2 & - 11 \end{bmatrix} }$$Now $\left( A^{15} + B \right)\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$
$${\Rightarrow \begin{bmatrix} 2 & - 11 \\ 2 & - 11 \end{bmatrix}\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} }{\Rightarrow 2x - 11y = 0}$$
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