n-th power of a matrix using the division of polynomials.

Question

Consider the matrix $$ A=\begin{pmatrix} 0 & 0 & 0\\ -2 & 1 & -1\\ 2 & 0 & 2 \end{pmatrix} $$

1. Calculate $A^3-3A^2+2A$.
2. What is the remainder of the division of the polynomial $X^n$ by the polynomial $X^3-3X^2+2X$.
3. Calculate $A^n$ for every natural number $n$.

I was solving the following problem and I was stuck in it. For part 1) the answer was the zero matrix. In part 2) I use the usual division and i get the following $$ X^n=X^{n-3}(X^3-3X^2+2X)+3X^{n-1}-2X^{n-2}. $$ When I pass to part 3) and using part 1) and 2), we obtain $$ A^n=3A^{n-1}-2A^{n-2}. $$ Using the fact that $A^3-3A^2+2A=O_{3\times 3}$. but if I use this answer for calculating $A^2$ the answer is not correct, so I think $A^n$ obtained is not correct. Now, one can use the diagonalization of the matrix $A$ and obtain $$ A^n=\begin{pmatrix} 0 & 0 & 0\\ -2^n & 1 & 1-2^n\\ 2^n & 0 & 2^n \end{pmatrix} $$

Can you help me in proving part 2 (if not correct) and part 3 without using the diagonalization method.

Answer 1

Write $$x^n = q(x)(x^3-3x^2+2x) + r(x) = q(x)x(x-1)(x-2)+r(x)$$ for polynomials $q,r \in \Bbb{R}[x]$ where $\deg r \le 2$. Plugging in $x = 0,1,2$ gives $$r(0) = 0, \quad r(1)=1, \quad r(2)=2^n$$ so $r(x) = (2^{n-1}-1)x^2+(-2^{n-1}+2)x$. Now we get $$A^n = q(A)(A^3-3A^2+2A) + r(A) = r(A) = (2^{n-1}-1)A^2+(-2^{n-1}+2)A$$ which yields precisely your result.

Answer 2

You could use induction.

For $n=1$, your statement is true.

Assuming $$ A^n=\begin{pmatrix} 0 & 0 & 0\\ -2^n & 1 & 1-2^n\\ 2^n & 0 & 2^n \end{pmatrix} $$

Then $$ A^{n+1} = AA^n = \begin{pmatrix} 0 & 0 & 0\\ -2 & 1 & -1\\ 2 & 0 & 2 \end{pmatrix} \begin{pmatrix} 0 & 0 & 0\\ -2^n & 1 & 1-2^n\\ 2^n & 0 & 2^n \end{pmatrix} = \begin{pmatrix} 0 & 0 & 0\\ -2^n - 2^n & 1 & 1-2^n - 2^n\\ 2\cdot2^n & 0 & 2\cdot2^n \end{pmatrix} = \begin{pmatrix} 0 & 0 & 0\\ -2^{n+1} & 1 & 1-2^{n+1}\\ 2^{n+1} & 0 & 2^{n+1} \end{pmatrix} $$

So $$ A^n = \begin{pmatrix} 0 & 0 & 0\\ -2^n & 1 & 1-2^n\\ 2^n & 0 & 2^n \end{pmatrix} $$

Is true for every natural number $n$