Let $\;A\;$ be a $\;2\times 2-$matrix with only one eigenvalue $\;x=5.\;$ Show that $\;(5I −A)^2 = 0.$

Question

I know that every matrix is conjugate to an upper triangle form matrix and conjugate matrices have the same characteristic polynomial.
I then try to get the characteristic polynomial of the upper triangle form and find it to be zero, however am unsure where to go from there.

Answer 1

The characteristic polynomial has only one root, namely $5$. What does this tell you about the polynomial? The only polynomials of degree $2$ with $5$ as the only root are of the form $p(x)=c(x-5)^{2}$ right? Every square matrix satisfies its characteristic polynomial and that should give you the answer.