# Definition The determinant of a [[square matrix]] is defined by: $\det(A) = \left(\sum_{j=1}^n a_{ij} a^{ij}\right) \quad \text{for any fixed} \, i $ # Properties $ \begin{align} \det(AB) &= \det(A) \det(B), \\ \det(B^{-1} A B) &= \det(A), \\ \det(A^T) &= \det(A),\\ \det(A) &= \lambda_1 \lambda_2 \ldots \lambda_n \\ \end{align} $ where $\lambda_i$ are the eigenvalues. Moreover, we also have: $ \det(e^X) = e^{\text{tr}(X)} $ where $\text{tr}$ denotes the [[trace]]. This identity is proved [[Trace and Determinant as Homomorphisms|here]].