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

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.

• Welcome, Rick! Think about the Cayley-Hamilton theorem! – Jose Brox Jan 22 at 9:27

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.