ThmDex – An index of mathematical definitions, results, and conjectures.
 ▼ Set of symbols ▼ Alphabet ▼ Deduction system ▼ Theory ▼ Zermelo-Fraenkel set theory ▼ Set ▼ Binary cartesian set product ▼ Binary relation ▼ Map ▼ Countable map ▼ Array ▼ Matrix ▼ Square matrix ▼ Set of square matrices ▼ Matrix determinant ▼ Complex matrix determinant
Definition D5718
Real matrix determinant

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 $$\text{Det} A : = \sum_{\pi \in S_N} \left( \text{Sign}(\pi) \prod_{n = 1}^N A_{n, \pi(n)} \right)$$

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 $$\text{Det} A : = \sum_{\pi \in S_N} \text{Sign}(\pi) A_{1, \pi(1)} A_{2, \pi(2)} A_{3, \pi(3)} \cdots A_{N - 1, \pi(N - 1)} A_{N, \pi(N)}$$
Results
 ▶ R5530: Determinant of a scaled real matrix ▶ R5511: Interchanging two rows or two columns of a real square matrix switches the sign of the determinant ▶ R5519: Real arithmetic expression for the determinant of a 2-by-2 real square matrix ▶ R5513: Real matrix determinant is homogeneous with respect to multiplying a row or a column by a constant ▶ R5510: Real square matrix which has a zero column or a zero row has determinant zero