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

Cofactor partition for the determinant of a real square matrix

Formulation 0

Let $A \in \mathbb{R}^{N \times N}$ be a D6160: Real square matrix such that

(i)	$N \in \{ 2, 3, 4, \ldots \}$ is a D5094: Positive integer
(ii)	$C_{i, j}$ is a D5941: Real square matrix cofactor for $A$ with respect to $(i, j)$ for each $i, j \in \{ 1, \ldots, N \}$

Let $m \in \{ 1, 2, \ldots, N \}$ be a D5094: Positive integer.

Then

(1)	\begin{equation} \text{Det} A = \sum_{n = 1}^N A_{m, n} C_{m, n} \end{equation}
(2)	\begin{equation} \text{Det} A = \sum_{n = 1}^N A_{n, m} C_{n, m} \end{equation}

Formulation 1

Let $A \in \mathbb{R}^{N \times N}$ be a D6160: Real square matrix such that

(i)	$N \in \{ 2, 3, 4, \ldots \}$ is a D5094: Positive integer
(ii)	$\text{Cof}_A(i, j)$ is a D5941: Real square matrix cofactor for $A$ with respect to $(i, j)$ for each $i, j \in \{ 1, \ldots, N \}$

Let $m \in \{ 1, 2, \ldots, N \}$ be a D5094: Positive integer.

Then

(1)	\begin{equation} \text{Det} A = \sum_{n = 1}^N A_{m, n} \text{Cof}_A(m, n) \end{equation}
(2)	\begin{equation} \text{Det} A = \sum_{n = 1}^N A_{n, m} \text{Cof}_A(n, m) \end{equation}

Proofs

Proof 0

Let $A \in \mathbb{R}^{N \times N}$ be a D6160: Real square matrix such that

(i)	$N \in \{ 2, 3, 4, \ldots \}$ is a D5094: Positive integer
(ii)	$C_{i, j}$ is a D5941: Real square matrix cofactor for $A$ with respect to $(i, j)$ for each $i, j \in \{ 1, \ldots, N \}$

Let $m \in \{ 1, 2, \ldots, N \}$ be a D5094: Positive integer.

This result is a particular case of R5064: Cofactor partition for a real square matrix. $\square$