Summary:

This section concerns the multiplicative inverse of square matrices. Given a square matrix A, we seek another square matrix B such that their product is an identity: AB = I. Thus, B is the inverse of A (and conversely, A is the inverse of B). Not every square matrix is invertible, and so we start to develop a rich theory of when such matrices are invertible.

We define the class of elementary matrices that are obtained by performing exactly one elementary row operation on an identity matrix. The elementary matrices are invertible, and moreover, left multiplication by an elementary matrix is the same as performing an elementary row operation.

This property is leveraged to prove one of our more important theorems of the semester: a matrix A is invertible if and only if its reduced row echelon form is the identity. This simultaneously gives a criteria for checking invertibility, and also an effective algorithm for computing the inverse: form the "superaugmented matrix" [A | I] and reduce to reduced row echelon form.

Outline:

Define the inverse of a square matrix.
Define the elementary matrices
Show that performing an elementary operation is the same as multiplying by an elementary matrix
Use the Gaussian elimination process to efficiently compute the inverse of a matrix.

Video lecture:

Part 1:

Part 2:

Part 3:

Notes from the lecture:

14.2 Section 2.2 The Inverse of a Matrix Part 1.pdf
14.2 Section 2.2 The Inverse of a Matrix Part 2.pdf
14.2 Section 2.2 The Inverse of a Matrix Part 3.pdf