# Determinant of a matrix ## Properties of matrix determinants 1. Invariance under [basis transformation](Change%20of%20basis.md) 2. The determinant is a [continuous function.](Continuous%20function.md) ^ea894a ## The determinant in $\mbox{GL}(n,\mathbb{F})$ The determinant is a [homomorphism](Homomorphism.md) of [$\mbox{GL}(n,\mathbb{F})$](GL(n;F).md) into $\mathbb{C}^*$. Here $\mathbb{C}^*$ is the set of complex numbers excluding $0$ since a determinant of $0$ implies that a given matrix is non-[invertible](Operator%20inverse.md) and thus not in $\mbox{GL}(n,\mathbb{F}).$ ## Evaluating the determinant of a matrix ### Determinants of block-diagonal matrices The determinant of a [block diagonal matrix](Block%20matrices.md#Block%20diagonal%20matrices) is the [product](Linear%20Algebra%20and%20Matrix%20Theory%20(index).md#Matrix%20multiplication) of its _[submatrices.](Block%20matrices.md)_ --- # Proofs and examples ## Proof that the determinant is a continuous function %% As a starting point check the validity of this method shown here: https://math.stackexchange.com/questions/1314411/proof-that-determinant-is-continuous-using-epsilon-delta-definition %% #MathematicalFoundations/Algebra/AbstractAlgebra/LinearAlgebra/Matrices