Matrix Proof: det(exp A) = exp(Tr A)

3-Second Blueprint (English)

1. Defining exp(A) via its power series and writing A = PJP^-1 in Jordan canonical form yields exp(A) = P exp(J) P^-1 for upper triangular J.
2. Determinant multiplicativity and trace similarity invariance cancel P and P^-1, reducing the proof to upper triangular matrices alone.
3. Since upper triangular powers multiply diagonal entries, exp(J) has diagonal e^λ_i whose product det(exp J) equals e^Σλ_i = exp(Tr A).