So I am asked to compute the power of a matrix using eigenvalues. Here is the question.
You are given that the vectors $p_1=(1,1,1), p_2=(2,1,1)$ and $p_3=(1,1,2)$ are eigenvectors to the matrix $$A=\begin{bmatrix} 3 &-4 & 2\\ 1 &-2 & 2\\ 1 &-5 & 5\\ \end{bmatrix}$$ a) Find the corresponding eigen values.
b) Explain why $B=(p_1,p_2,p_3)$ is an ordered basis of $\mathbb{R}^3$.
c) Find the coordinate vector $[v]_B$, relative to $B$, if $v=(1,1,0)$.
d) Find $A^{2020}v$.
a) I already found the eigenvalues. They're $1$,$2$ and $3$.
b) It's an ordered basis because the eigenvalues are distict so that means the vectors are linearly independent. The determinant test also confirms this. So they form a valid basis.
c) $[v]_B=<2,0,-1>$.
d) I'm pretty sure I have to diagonalize the matrix here. So that means I get $D=\begin{bmatrix} 1 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 3\\ \end{bmatrix}$, $P=\begin{bmatrix} 1 & 2 & 1\\ 1 & 1 & 1\\ 1 & 1 & 2\\ \end{bmatrix}$, and $P^{-1}=\begin{bmatrix} -1 & 3 & -1\\ 1 & -1 & 0\\ 0 & -1 & 1\\ \end{bmatrix}$
This means that $A^{2020}=PD^{2020}P^{-1}$
My problem is that $D^{2020}$ and the subsequent multiplication is so unbelivably large, I am not sure if what I am doing is actually correct.
Is my procedure correct or am I doing something fundamentally wrong?