A two-factor hedge uses independent exposure vectors to span the target risk. The determinant measures the area or volume those vectors span in their chosen units; Cramer's rule expresses exact holdings as ratios of such volumes. This note derives the product and inverse rules and shows a small exposure solve. For larger systems, LU factorisation solves directly without computing a separate determinant for each holding.
Learning objectives. By the end of this note you should be able to:
- interpret as a volume and explain why means the spanning vectors are degenerate;
- state the product rule, derive the inverse rule from it, and prove transpose invariance using the permutation formula;
- apply Cramer's rule to solve a small system;
- explain why Cramer's rule, despite its elegance, is not the practical method.
In this lesson
- Determinant as volume
- The multiplicative property
- Cramer's rule
Before you start
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