Calculate similarity of symbolic matrix and power of some variable of that symbolic matrix

Question

I have the matrices

$ A=\left( \begin{matrix} α & 0 & 0 \\ 0 & α & 0 \\ 0 & 0 & α \end{matrix} \right)$ and $ N=\left( \begin{matrix} 0 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix} \right)$

I need to prove that $AN=NA$, and then calculate $(A+N)^k$

Do I have to use characteristic polynominal to proof it ?

For matrix N, I have calculated : $det(N) = 0$, $ Eigenvalue = 0$, $ Eigenvector = [1,0,0]$

I'm not sure how to calculate with the matrix A, I need proof this manually (not using software). And as for the power of k, how do you calculate matrix with the power of some variable when the matrix itself is symbolic ?

Thanks !

Answer 1

No, you don't need characteristic polynomials.

Hint 1. Your matrix $A$ could be written shorter as $\alpha I$, and scalar multiples of the identity matrix commute with everything!

Hint 2. There is a simple pattern to the powers of your $N$. Compute the first few powers by hand, and it will be impossible for you not to notice it.

Hint 3. Since $A$ and $N$ commute, the binomial theorem applies to $(A+N)^k$.

Hint 4. After you have combined hints 2 and 3, collect like terms. Notice that the big complex coefficient is almost equal to the binomial expansion of $(\alpha+1)^k$.