Real matrix determinant

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Real matrix determinant

Formulation 0

Let $A \in \mathbb{R}^{N \times N}$ be a D6160: Real square matrix.
Let $S_N$ be a D4951: Set of standard N-permutations.
The determinant of $A$ is the D993: Real number \begin{equation} \text{Det} A : = \sum_{\pi \in S_N} \left( \text{Sign}(\pi) \prod_{n = 1}^N A_{n, \pi(n)} \right) \end{equation}

Formulation 1

Let $A \in \mathbb{R}^{N \times N}$ be a D6160: Real square matrix.
Let $S_N$ be a D4951: Set of standard N-permutations.
The determinant of $A$ is the D993: Real number \begin{equation} \text{Det} A : = \sum_{\pi \in S_N} \text{Sign}(\pi) A_{1, \pi(1)} A_{2, \pi(2)} A_{3, \pi(3)} \cdots A_{N - 1, \pi(N - 1)} A_{N, \pi(N)} \end{equation}

Results

» R2466: Real matrix determinant is multiplicative

» R5064: Cofactor partition for a real square matrix

» R5511: Interchanging two rows or two columns of a real square matrix switches the sign of the determinant

» R5512: Determinant zero if a real square matrix contains two identical rows or two identical columns

» R5519: Real arithmetic expression for the determinant of a 2-by-2 real square matrix

» R5510: Real square matrix which has a zero column or a zero row has determinant zero

» R5513: Real matrix determinant is homogeneous with respect to multiplying a row or a column by a constant

» R5530: Determinant of a scaled real matrix