If A is a $2×2 $ matrix, does $A^2=3A$ imply $ A=0$ or $A=3I$? Problem: Let $A$ be a $2\times 2$ matrix, $O$ be the null matrix, and $I$ be the identity matrix. Is the following statement true?
$$\text{if } A^2=3A \implies A=O\text{ or }3I$$
I tried proving this just by factorising $A^2-3A=O \implies A(A-3I)=O \implies A-3I=O, A=O$. However, I cannot tell if this is correct. Do polynomials work the same way with matricies as they do with real/complex numbers?

As soon as a polynomial $P$ has two distinct roots $\alpha,\beta$, the polynomial equation $P[A]=0$ for $A$ a $2\times2$ matrix has "mixed" solutions
$$
  A=\pmatrix{\alpha&0\\0&\beta}
\quad\text{and}\quad
  A=\pmatrix{\beta&0\\0&\alpha}
$$
(because doing arithmetic with diagonal matrices basically means doing that arithmetic in each diagonal position separately, while leaving the off-diagonal positions zero), as well as "pure" solutions $A=\alpha I$ and $A=\beta I$.
moreover, the number of solutions is not even finite (if the field of scalars is infinite), because one can apply any change of basis $A\mapsto P^{-1}AP$ to the mixed solutions to get infinitely many different ones. (The pure solutions do not change under change of basis.)
So the answer is no, not just those two solutions.
