Unit 5 of 11
Determinants
- 3 lessons
- Videos 24 to 26
The determinant of a square matrix is the signed factor by which its map scales area or volume. This unit computes determinants and uses them to test invertibility.
The sign records whether the map keeps or reverses orientation. A zero determinant means the columns are dependent and the map flattens space onto a lower-dimensional subspace. Row operations, triangular form and cofactor expansion compute determinants. The determinant of a product is the product of the determinants, which matches composing the maps, and Cramer's rule solves small square systems whose coefficient matrix has a nonzero determinant.
Every lesson will be one video. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.
Area, Volume and Orientation
Determinants as Oriented Area and Volume
How the determinant measures oriented area and volume, why a 2 by 2 determinant is ad minus bc, how its sign records orientation, and why a zero determinant means dependent columns.
Video 24
Computing and Using Determinants
Computing Determinants with Row Operations
How each row operation changes a determinant, why a triangular determinant is the product of its diagonal, how minors and cofactors give cofactor expansion, and why the determinant is linear in one column and multiplicative.
Video 25
Determinants, Invertibility and Cramer's Rule
Why a square matrix is invertible exactly when its determinant is nonzero, which right-hand sides a singular square system can reach, how Cramer's rule solves a nonsingular system, and when elimination is the better method.
Video 26