ThmDex – An index of mathematical definitions, results, and conjectures.
F13085
Formulation 0
Let $A \in \mathbb{C}^{N \times N}$ be a D6159: Complex square matrix such that
(i) $a_1, \ldots, a_N \in \mathbb{C}^{N \times 1}$ are each a D5689: Complex column matrix
(ii) $b_1, \ldots, b_N \in \mathbb{C}^{1 \times N}$ are each a D5688: Complex row matrix
(iii) \begin{equation} A = \begin{bmatrix} a_1 & a_2 & \cdots & a_N \end{bmatrix} \end{equation}
(iv) \begin{equation} A = \begin{bmatrix} b_1 \\ b_2 \\ \vdots \\ b_N \end{bmatrix} \end{equation}
Then
(1) \begin{equation} a_1 = \boldsymbol{0} \text{ or } a_2 = \boldsymbol{0} \text{ or } \cdots \text{ or } a_N = \boldsymbol{0} \quad \implies \quad \text{Det} A = 0 \end{equation}
(2) \begin{equation} b_1 = \boldsymbol{0} \text{ or } b_2 = \boldsymbol{0} \text{ or } \cdots \text{ or } b_N = \boldsymbol{0} \quad \implies \quad \text{Det} A = 0 \end{equation}