Let $A ∈ M_{2×2}(\mathbb{C})$ be a matrix having a unique eigenvalue $c$. Show that $A^n = c^{n−1}[nA − (n − 1)cI ]$ for all $n > 0$.

Question

Let $A ∈ M_{2×2}(\mathbb{C})$ be a matrix having a unique eigenvalue $c$. Show that $A^n = c^{n−1}[nA − (n − 1)cI ]$ for all $n > 0$.

I'm doing induction for this, the base step when $n=1$ gives us $A=A$. Now, assume it's true for some $k$ and consider $A^{k+1}=A^k A=c^{k−1}[kA − (k − 1)cI ]A$. I'm stuck trying to figure this out. Any solutions/help is greatly appreciated.

Answer 1

Hint: Since $A$ has a unique eigenvalue $c$, its characteristic polynomial is $p(\lambda) = (\lambda - c)^2$. By the Cayley-Hamilton theorem, this implies $(A-cI)^2 = 0 \implies A^2 = 2cA - c^2 I$. Use this fact to complete the inductive step.