This is similar to Dan Petersen's answer, but more elementary. A fact I always mention when talking to students about matrix groups is that the diagonalisable matrices (over $\mathbb{C}$) are dense in the space of all matrices.

This means that various things can be checked just on the diagonalisable matrices. For example, to establish the formula $\det(\exp(A)) = e^{\mathrm{tr}(A)}$ for $A$ a complex-valued matrix: observe that both sides define continuous functions $M_n(\mathbb{C}) \to \mathbb{C}$, and they clearly agree on the diagonal matrices, hence they also agree on the diagonalisable matrices (as $\det$ and $\mathrm{tr}$ are conjugation-invariant), so by continuity they agree on all matrices.

Of course, this example contains within it another application on the principle in question: the proof does not work if we insist on staying within matrices over $\mathbb{R}$ (the diagonalisable matrices are no longer dense), so even to prove the result over $\mathbb{R}$ it is easier to generalise to $\mathbb{C}$ first.