Find a $3 \times 3$ matrix $A$ whose minimal polynomial is $m_A(x) = x^2$.
From the Cayley-Hamilton theorem, $m_A(x) = 0 \implies A^2 = 0$. So, A is nilponent so, its characteristic polynomial is $P_A(x)=x^3$,since $m_A(x)=x^2$. That means that $A \neq0$. I don't know what to do from here. Any tips?
Best Answer
I'd try a matrix with zeroes everywhere except in one entry (this one, not in the main diagonal).