How to prove that $\operatorname{Tr}(A)=a+b$ and $\det(A)=ab$ if characteristic polynomial of $A$ is degreed 2 and with $a$ and $b$ as eigenvalue?

If $a$ and $b$ are distinct eigenvalues, then $A$ is diagonalizable, so that $\operatorname{Tr}(A)=a+b$ and $\det(A)=ab$ since there's a similar diagonal matrix. However, how are we making sure that there are no zeros in $A$ as we don't know the size of $A$?