A tenth power of a matrix

MediumLinear Algebra~6m

Let $A = \begin{bmatrix} 3 & 1 \\ 4 & 3 \end{bmatrix}$ and $v = \begin{bmatrix} 3 \\ 2 \end{bmatrix}$ . Then $A^{10}v = \begin{bmatrix} a\cdot b^{p} + c \\ d\cdot b^{p} - g \end{bmatrix}$ for positive integers $a, b, c, d, p, g$ (with $b^p$ the same in both entries). Find $b\,p + a + c + d + g$ .

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Find the eigenvalues ( $5$ and $1$ ) and eigenvectors, then write $v$ in that basis.
$A^{10}$ scales each eigencomponent by $\lambda^{10}$ ; recombine to read off $a,b,p,c,d,g$ .

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A tenth power of a matrix

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