Determinants: Evaluation and Properties

Determinants: Evaluation and Properties

This lesson covers the following key concepts:

• Value of 2×2 and 3×3 determinants
• Minor and cofactor of elements
• Expansion along any row or column
• Properties: row/column operations, transpose, product
• Product of two determinants
• Differentiation and integration of determinants
• Special determinants: Vandermonde, circulant

Important Formulas

• $$\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$$
• $$M_{ij} = \text{minor}, A_{ij} = (-1)^{i+j}M_{ij} \text{ (cofactor)}$$
• $$|A^T| = |A|$$
• $$|AB| = |A| \cdot |B|$$
• $$|kA| = k^n|A| \text{ for } n \times n$$
• $$\text{Interchange rows: } |A| \rightarrow -|A|$$
• $$\text{Vandermonde: } (b-a)(c-a)(c-b)$$