3. Multiplication and Inverse Matrices

Overview

Gilbert Strang develops five equivalent ways to understand matrix multiplication: entry-by-entry, by columns, by rows, as sums of column-row products, and by blocks. He then connects invertibility to whether a square matrix has a nonzero vector in its nullspace and derives Gauss-Jordan elimination, which finds an inverse by reducing the augmented matrix [A|I] to [I|A⁻¹].

Key takeaways

For A of size m × n and B of size n × p, AB has size m × p, and each entry is cᵢⱼ = Σₖ aᵢₖbₖⱼ.
The columns of AB are combinations of A's columns, while its rows are combinations of B's rows; these views expose how multiplication transforms column and row spaces.
A product AB can also be written as a sum of outer products: column k of A times row k of B, summed over the shared dimension.
A square matrix with a nonzero vector x satisfying Ax = 0 cannot be invertible, because applying A⁻¹ would incorrectly force the nonzero x to equal zero.
Gauss-Jordan finds A⁻¹ by reducing [A|I] to [I|A⁻¹]; the row operations form a matrix E satisfying EA = I, so E = A⁻¹.
For A = [1 3; 2 7], the inverse is [7, -3; -2, 1], while [1 3; 2 6] is singular because [3, -1]ᵀ lies in its nullspace.

Chapters

0:00 Matrix Multiplication Entries and Compatible Dimensions

For C = AB, the entry cᵢⱼ is the dot product of row i of A with column j of B: cᵢⱼ = Σₖ aᵢₖbₖⱼ.
If A is m × n and B is n × p, their inner dimensions n must match, and C has shape m × p.
The example c₃₄ sums products from row 3 of A and column 4 of B, matching indices across the shared dimension.

6:06 Computing AB One Column at a Time

Each column of C = AB is A multiplied by the corresponding column of B.
Because multiplying A by a vector forms a combination of A's columns, every column of C lies in the column space of A.
The coefficients of each combination are the entries in the corresponding column of B.

10:09 Row Combinations and Column-Times-Row Products

Each row of AB is a combination of the rows of B, with the coefficients determined by the corresponding row of A.
A column of length m times a row of length p produces an m × p matrix, not a scalar.
The outer product of [2, 3, 4]ᵀ and [1, 6] is a 3 × 2 matrix whose rows and columns are scalar multiples of the input vectors.
In general, AB is the sum over k of column k of A times row k of B.

18:18 Block Matrix Multiplication by Block Rows and Columns

Matrices can be partitioned into blocks and multiplied using the same row-by-column rule as scalar-entry matrices.
For a 2 × 2 block partition, the upper-left output block is A₁B₁ + A₂B₃.
Blocks may themselves be matrices, such as 10 × 10 blocks within a 20 × 20 matrix, as long as the block dimensions match.

21:22 Defining the Inverse of a Square Matrix

A square matrix A is invertible if a matrix A⁻¹ exists with A⁻¹A = I.
For square matrices, a left inverse is also a right inverse, so AA⁻¹ = I as well.
Rectangular matrices can have a left inverse without a right inverse because their dimensions constrain which products are defined.

24:09 Why the Matrix [1 3; 2 6] Has No Inverse

The rows and columns of [1 3; 2 6] are linearly dependent; its determinant is zero.
Multiplying this matrix by another matrix can only produce columns in the line spanned by [1, 2]ᵀ, so it cannot produce the identity's columns.
The nonzero vector [3, -1]ᵀ satisfies A x = 0 because 3 times the first column minus the second column equals zero.
If A had an inverse, multiplying A x = 0 by A⁻¹ would imply x = 0, contradicting [3, -1]ᵀ ≠ 0.

31:52 Finding the Inverse of [1 3; 2 7] by Solving Systems

The matrix A = [1 3; 2 7] has independent columns and determinant 1, so it is invertible.
Each column xⱼ of A⁻¹ solves A xⱼ = eⱼ, where eⱼ is the corresponding column of the identity.
For this 2 × 2 example, finding A⁻¹ means solving two systems with the same coefficient matrix and right-hand sides [1, 0]ᵀ and [0, 1]ᵀ.

36:13 Gauss-Jordan Elimination on the Augmented Matrix [A|I]

Gauss-Jordan solves for all columns of A⁻¹ at once by appending the identity to A.
For A = [1 3; 2 7], elimination changes [A|I] into [I|A⁻¹], yielding A⁻¹ = [7, -3; -2, 1].
The resulting inverse can be checked by multiplication: [7, -3; -2, 1][1, 3; 2, 7] = I.

42:20 Why Row Reduction Turns the Identity into A⁻¹

Every row operation is left multiplication by an elimination matrix; a sequence of operations is equivalent to multiplication by their product E.
Reducing the left half of [A|I] to I means EA = I, so E is A⁻¹.
The same E multiplies the identity on the right half, turning I into A⁻¹ and explaining the Gauss-Jordan procedure.
For an n × n matrix, the augmented identity supplies n right-hand sides, producing all n columns of the inverse in one elimination process.

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