ThmDex – An index of mathematical definitions, results, and conjectures.
Result R5517 on D5717: Complex matrix determinant
Cofactor partition for a complex square matrix
Formulation 0
Let $A \in \mathbb{C}^{N \times N}$ be a D6159: Complex square matrix such that
(i) $N \in \{ 2, 3, 4, \ldots \}$ is a D5094: Positive integer
(ii) $C_{i, j}$ is a D6183: Complex square matrix cofactor for $A$ with respect to $(i, j)$ for each $i, j \in \{ 1, \ldots, N \}$
Let $i, j \in \{ 1, 2, \ldots, N \}$ each be a D5094: Positive integer.
Then
(1) \begin{equation} i = j \quad \implies \quad \sum_{n = 1}^N A_{i, n} C_{j, n} = \text{Det} A \end{equation}
(2) \begin{equation} i = j \quad \implies \quad \sum_{n = 1}^N A_{n, i} C_{n, j} = \text{Det} A \end{equation}
(3) \begin{equation} i \neq j \quad \implies \quad \sum_{n = 1}^N A_{i, n} C_{j, n} = 0 \end{equation}
(4) \begin{equation} i \neq j \quad \implies \quad \sum_{n = 1}^N A_{n, i} C_{n, j} = 0 \end{equation}
Formulation 1
Let $A \in \mathbb{C}^{N \times N}$ be a D6159: Complex square matrix such that
(i) $N \in \{ 2, 3, 4, \ldots \}$ is a D5094: Positive integer
(ii) $C_{i, j}$ is a D6183: Complex square matrix cofactor for $A$ with respect to $(i, j)$ for each $i, j \in \{ 1, \ldots, N \}$
Let $i, j \in \{ 1, 2, \ldots, N \}$ each be a D5094: Positive integer.
Then
(1) \begin{equation} i = j \quad \implies \quad \sum_{n = 1}^N A_{i, n} \text{Cof}_A(j, n) = \text{Det} A \end{equation}
(2) \begin{equation} i = j \quad \implies \quad \sum_{n = 1}^N A_{n, i} \text{Cof}_A(n, j) = \text{Det} A \end{equation}
(3) \begin{equation} i \neq j \quad \implies \quad \sum_{n = 1}^N A_{i, n} \text{Cof}_A(j, n) = 0 \end{equation}
(4) \begin{equation} i \neq j \quad \implies \quad \sum_{n = 1}^N A_{n, i} \text{Cof}_A(n, j) = 0 \end{equation}
Also known as
Laplace expansion for a complex square matrix, Cofactor expansion for a complex square matrix
Subresults
▶ R5520: Cofactor partition for a 2-by-2 complex square matrix
▶ R5525: Complex square matrix which has a zero column or a zero row has determinant zero