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?

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*$A-3$ doesn't makes sense. We can add two matrix and we can multiply a matrix by a scalar( one at a time, known as linear combination).

$A-3$ should be $A-3I$ .


*$A(A-3I) =0$ doesn't imply $A=0$ or $A=3I$ . Cancellation doesn't holds.$M_2(F) $ is not an integral domain (contains zero divisors)

Counter example:
$A=\begin{pmatrix}0&0\\0&3\end{pmatrix}$
Then $A^2-3A=0$ but neither $A=0$ nor $A=3I$
