20. Cramer's Rule, Inverse Matrix, and Volume

Overview

The lecture derives the inverse formula A⁻¹ = Cᵀ/det(A), where C is the cofactor matrix, and uses it to obtain Cramer's Rule: each solution component xⱼ equals det(Bⱼ)/det(A), with column j of A replaced by b. It then interprets determinants geometrically as signed areas and volumes, using parallelepipeds, orthogonal matrices, and the 2D formula ad − bc to connect determinant properties with geometric scaling.

Key takeaways

For an invertible matrix A, the cofactor formula gives A⁻¹ = Cᵀ/det(A); the transpose places each cofactor in the position needed for A Cᵀ to equal det(A)I.
Cramer's Rule follows from the inverse formula: xⱼ = det(Bⱼ)/det(A), where Bⱼ replaces column j of A with the right-hand side b.
Cramer's Rule is valuable as a symbolic formula but inefficient for computation because it requires n + 1 determinants; Gaussian elimination is the practical approach.
The absolute determinant is the volume of the parallelepiped formed by a matrix's vectors, while its sign records orientation.
Orthogonal matrices preserve the unit cube's volume because QᵀQ = I implies det(Q)² = 1.
In two dimensions, the determinant ad − bc gives parallelogram area, and half its absolute value gives the area of the triangle formed with the origin.

Chapters

0:00 The Cofactor Formula for a Matrix Inverse

The 2×2 inverse formula divides by det(A), matching the condition that A is invertible exactly when det(A) is nonzero.
For a 2×2 matrix, the inverse entries d, −b, −c, and a correspond to cofactors arranged in transposed positions.
The general candidate is A⁻¹ = Cᵀ/det(A), where C contains the cofactors of A.

7:30 Verifying the Adjugate Identity with Cofactor Expansion

The inverse formula is verified by checking A Cᵀ = det(A)I.
Each diagonal entry of A Cᵀ is a row of A multiplied by the cofactors from that same row, giving det(A) by cofactor expansion.
Each off-diagonal entry is zero because it corresponds to a determinant with two identical rows.
The determinant uses products of n matrix entries, while each cofactor uses products of n − 1 entries.

18:52 Cramer's Rule from the Inverse Formula

For Ax = b, the unique solution is x = A⁻¹b when det(A) is nonzero.
The jth component of Cᵀb is the determinant of Bⱼ, formed by replacing column j of A with b.
Cramer's Rule gives xⱼ = det(Bⱼ)/det(A), expressing every solution component as a ratio of determinants.

25:30 Why Cramer's Rule Is a Poor Numerical Method

Computing a solution through Cramer's Rule requires finding det(A) and n additional determinants.
For practical numerical work, Gaussian elimination is far more efficient than repeatedly computing determinants.
The explicit inverse and Cramer's Rule remain useful as algebraic formulas, but software such as MATLAB uses elimination rather than Cramer's Rule.

28:06 A Matrix's Rows Define a Parallelepiped

For a 3×3 matrix, its three row vectors can be drawn as edges from the origin to define a parallelepiped.
The determinant gives the shape's signed volume, while the ordinary nonnegative volume is |det(A)|.
Changing the sign of the determinant distinguishes orientation, such as a right-handed versus left-handed arrangement of edges.

34:07 Identity and Orthogonal Matrices Preserve Unit Volume

The identity matrix defines the unit cube, whose volume and determinant are both 1.
An orthogonal matrix Q has perpendicular unit-length columns, so it rotates or reflects the unit cube without changing its volume.
Taking determinants of QᵀQ = I gives det(Q)² = 1, so det(Q) is 1 or −1, consistent with unit volume and orientation.

38:38 Determinant Scaling and the Volume Argument

Doubling one edge of a rectangular box doubles its volume.
Doubling a row of a matrix doubles its determinant, by the determinant's row-scaling property.
The comparison with determinant properties builds the proof that determinant magnitude measures volume, including the need to account for row addition and non-right angles.

43:30 The 2D Determinant Gives Parallelogram Area

Two vectors (a, b) and (c, d) define a parallelogram in the plane.
Its signed area is ad − bc, the determinant of the matrix with those vectors as rows.
This coordinate formula avoids separately finding a base length, a perpendicular height, or angles.

46:37 Triangle Area from Two Vectors at the Origin

A triangle with vertices (0, 0), (a, b), and (c, d) occupies half the parallelogram formed by the two nonzero vertices.
Its area is ½|ad − bc|, using the same 2×2 determinant as the parallelogram formula.
The determinant method is especially useful when vertex coordinates are known but base lengths and heights are not.

48:15 The Determinant Formula for Any Triangle's Vertices

For arbitrary vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃), the signed area is half the determinant of the matrix with rows (xᵢ, yᵢ, 1).
Subtracting one vertex row from the other two reduces the expression to a triangle based at the origin, showing how the formula encodes translation.
The lecture concludes with determinants as a compact coordinate-based formula for area and volume, before moving on to eigenvalues.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, MIT OpenCourseWare.