The previous note gave cofactor expansion as a recursive computation method without showing where the checkerboard sign pattern ultimately comes from. This sub-note derives the fully explicit permutation formula — a single sum over every way of picking one entry from each row and column — starting from the three defining properties of the first note in this topic, and shows cofactor expansion is exactly this sum, grouped by which entry of one chosen row is picked.
Learning objectives. By the end of this note you should be able to:
- expand a determinant fully into a sum of six signed products, by row-splitting;
- state the general permutation formula and identify the role of a permutation's sign;
- connect the permutation formula to cofactor expansion by grouping terms;
- explain why this formula, while completely explicit, is never used for actual computation.
In this lesson
- Splitting every row
- The full formula
- Cofactor expansion is this formula, regrouped
- Why this formula is never used for computation
Before you start
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