Determinant

Dependencies:

1. Ring
2. Submatrix

Let $A$ be an $n$ by $n$ matrix over a ring with unity.

The determinant of $A$, denoted as $\operatorname{det}(A)$ or $|A|$, is defined recursively:

\[ \operatorname{det}(A) = \begin{cases} A[1, 1] & n = 1 \\ \sum_{i=1}^n (-1)^{n+i}A[n, i]\operatorname{det}(A[-n, -i]) & n > 1 \end{cases} \]

Dependency for:

1. Determinant after elementary row operation
2. Swapping last 2 rows of a matrix negates its determinant
3. Determinant of upper triangular matrix

Info:

• Depth: 4
• Number of transitive dependencies: 5

Transitive dependencies:

1. Group
2. Ring
3. Semiring
4. Matrix
5. Submatrix