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This lecture introduces determinants — a single number that tells you whether a matrix is invertible and how it scales areas/volumes. The 2×2 formula $ad - bc$ is derived geometrically from the area of the transformed unit square, then generalised to $n \times n$ via cofactor expansion. Key properties (transpose, product, triangular, row operations) are covered, along with Cramer's rule and the crucial insight that determinants are invariant under basis transformations. Lecturer: Dr Markus Brede.

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📋 At a Glance

• [ ] Derive the 2×2 determinant $\det A = ad - bc$ from the area of the transformed unit square
• [ ] Explain why it's called "determinant": $\det A = 0$ ↔ $A$ is not invertible (columns dependent)
• [ ] Define the minor $M_{ij}$: determinant of the submatrix with row $i$ and column $j$ deleted
• [ ] Define the cofactor $C_{ij} = (-1)^{i+j} M_{ij}$
• [ ] Compute determinants via cofactor expansion along any row or column
• [ ] Verify cofactor expansion agrees with $ad - bc$ for the 2×2 case
• [ ] Work through the 3×3 example: $\det \begin{pmatrix} 1&2&3 \\ 3&2&1 \\ 2&1&3 \end{pmatrix} = -12$
• [ ] State and apply the 5 key properties (transpose, triangular, product, invertibility, inverse)
• [ ] State and apply the 3 row operation rules (swap → $times(-1)$, add rows → unchanged, scale row → $times b$)
• [ ] Compute determinants via row reduction to triangular form (the efficient method)
• [ ] Know the computational complexity: cofactor expansion = $O(n!)$, GE method = $O(n^3)$
• [ ] State Cramer's rule: $A^{-1} = \frac{1}{\det A} C^T$ (theoretically useful, computationally expensive)
• [ ] Prove that determinants are invariant under similarity transformations: $\det(M^{-1}AM) = \det A$

1. Motivation: What Do Linear Transformations Do to Area?

Linear transformations change the shape of the unit square — they can scale, rotate, reflect, or shear it. The key question: by what factor does the area change?

If we know how a transformation scales the unit square, we know how it scales any shape (approximate any region with tiny boxes).