Lecture 13: Introduction to Matrix Inverses (Nicholson Section 2.4)
Sub Project: Linear Algebra Lecture Videos
Alternate Video Access via MyMedia | Video Duration: 47:53
Description: Started by reminding students that Ax=b will have a) no solution, b) exactly one solution, or c) infinitely many solutions and discussed what this had to do with the Column Space of A and rank(A). If you don’t know what the Column Space of A is yet, ignore that part!
4:40 --- one option to trying to solve Ax=b is via elementary row operations. Discussed the costs & benefits of this approach.
6:20 --- another option is to find a matrix B (if it can be found) so that AB = I (the identity matrix) and use B to find the solution x. Discussed the costs & benefits of this approach.
12:50 --- when is it better to use elementary row operations to try and solve Ax=b and when is it better to try and find B so that BA=I?
16:00 --- does every square matrix have some matrix so that BA = I? Gave an example of a 2x2 matrix for which there is no B so that BA=I --- presented two different arguments as to why there could never be a B so that BA=I.
25:00 --- presented a super-important and super-useful theorem about matrix inverses.
31:20 --- used theorem to construct an algorithm to try and find a matrix B so that BA=I. Note: the algorithm is using block multiplication. Specifically, if A is 2x2 and B is 2x2 with B = [b1;b2] then AB = A [b1; b2] = [Ab1;A b2] = [e1; e2] = I. Make sure that you’re comfortable with this block multiplication!
37:45 --- Used the matrix inversion algorithm on a 2x2 matrix for which there is a B so that BA=I.
40:50 --- Used the matrix inversion algorithm on a 2x2 matrix for which there isn’t a B so that BA=I.
43:30 --- bird’s eye view of matrix inversion algorithm.
46:20 --- Defined what it means for a square matrix to be invertible.
