Consider the Difference Equation $\mathbf { x } _ { \mathbf { k } + 1 } = \mathrm { Ax } _ { \mathbf { k } }$

Consider the difference equation $\mathbf { x } _ { \mathbf { k } + 1 } = \mathrm { Ax } _ { \mathbf { k } }$ , where $\mathrm { A }$ has eigenvalues and corresponding eigenvectors $\mathbf { v } _ { 1 }$ , $\mathbf { v } _ { 2 }$ , and $\mathbf { v } _ { 3 }$ given below. Find the general solution of this difference equation if $x _ { 0 }$ is given as below.

- $\lambda _ { 1 } = 1.3 , \lambda _ { 2 } = 0.8 , \lambda _ { 3 } = 0.6 , \mathbf { v } _ { 1 } = \left[ \begin{array} { r } - 6 \\ 6 \\ 1 \end{array} \right] , \mathbf { v } _ { 2 } = \left[ \begin{array} { l } 2 \\ 1 \\ 2 \end{array} \right] , \mathbf { v } _ { 3 } = \left[ \begin{array} { r } 1 \\ 2 \\ - 2 \end{array} \right]$ , and $\mathbf { x } _ { 0 } = \left[ \begin{array} { r } - 33 \\ 33 \\ - 7 \end{array} \right]$

A)
$\mathrm{x}_{\mathrm{k}}=(1.3) ^{\mathrm{k}_{\mathbf{v1}}}-3(0.8) ^{\mathrm{k}_{\mathbf{}}} \mathbf{v}_{2}+3(0.6) ^{\mathrm{k}_{\mathbf{v3}}}$
B)
$\mathrm{x}_{\mathrm{k}}=(1.3) ^{\mathrm{k}_{\mathbf{v1}}}+(0.8) ^{\mathrm{k}_{\mathbf{v2}}}+(0.6) ^{\mathrm{k}_{\mathbf{v3}}}$


C) $\mathbf { x } _ { \mathrm { k } } = 5 ( 1.3 ) ^ { \mathrm { k } _ { \mathbf { v1 } } } - 3 ( 0.8 ) ^ { \mathrm { k } _ { \mathbf {v2} } } + ( 0.6 ) ^ { \mathrm { k } _ { \mathbf { v3 } } }$

D) $\mathbf { x } _ { \mathrm { k } } = 5 ( 1.3 ) ^ { \mathrm { k } _ { \mathbf { v1 } } } - 3 ( 0.8 ) ^ { \mathrm { k } _ { \mathbf { v2 } } } + 3 ( 0.6 ) ^ { \mathrm { k } _ { \mathbf { v3 } } }$