What you will learn in Matrices & Determinants

1. 1. Matrices, Order and Types

   An array, a table, or a transformation — and the names that overlap on purpose.

2. 2. Addition and Scalar Multiplication

   The easy operations, and exactly when they are defined.

3. 3. Matrix Multiplication

   The row–column rule, why the dimensions must agree, and why the order matters.

4. 4. Matrices as Geometric Transformations

   The module's central picture, arriving before the determinant needs it.

5. 5. Transpose, Symmetric and Skew-Symmetric Matrices

   Reflecting in the diagonal, and the split every square matrix has.

6. 6. The 2×2 Determinant as Signed Area

   ad − bc, and what those four products are measuring.

7. 7. The 3×3 Determinant and Cofactor Expansion

   The same idea one dimension up, where the area becomes a volume.

8. 8. Properties of Determinants

   Three row operations with stated effects, and one very natural false statement.

9. 9. Minors, Cofactors and the Adjoint

   Bookkeeping, made systematic enough to build an inverse out of.

10. 10. The Inverse of a Matrix

    Undoing a transformation, and finding out when there is no undo.

11. 11. Solving Linear Systems with Matrices

    Equations, geometry and a matrix equation — and the two cases a zero determinant cannot tell apart.

12. 12. Cramer's Rule, Consistency and Rank

    A ratio of determinants with a reason behind it, and the tool that classifies every case.