ThmDex – An index of mathematical definitions, results, and conjectures.

Result R5080 on D5717: Complex matrix determinant

Complex matrix determinant nonzero iff kernel is trivial

Formulation 0

Let $A \in \mathbb{C}^{N \times N}$ be a D999: Complex matrix.

Then the following statements are equivalent

(1)	\begin{equation} \text{det}(A) \neq 0 \end{equation}
(2)	\begin{equation} \text{Ker}(A) = \left\{ (0, 0, \ldots, 0)^T \right\} \end{equation}

Formulation 1

Let $A \in \mathbb{C}^{N \times N}$ be a D999: Complex matrix.

Then the following statements are equivalent

(1)	\begin{equation} \text{det}(A) \neq 0 \end{equation}
(2)	\begin{equation} \left\{ z \in \mathbb{C}^{N \times 1} : A z = \boldsymbol{0} \right\} = \left\{ \boldsymbol{0} \right\} \end{equation}