Calculating Determinants

Last updated Mar 6, 2022 Edit Source

• Math linear

Calculating determinants with the concept of cofactor is relatively straight forward. Given an $n \times m$ matrix $A$, the determinant is defined by one of the following formulas:

$$ \forall k \leq m : det(A) = \sum_{i=0}^n -1^{i+k}a_{ik}C_{ik} $$

$$ \forall k \leq n : det(A) = \sum_{i=0}^m -1^{i+k}a_{ki}C_{ki} $$

Notice that to calculate the determinant, we can pick any row/column we want, so it is best to pick one with as many zeros as possible.

• $det(A^T) = det(A)$
• $det(AB) = det(A)det(B)$
• $det(A^{-1}) = det(\frac{1}{A})$