Lecture 14: Properties of Inverse Matrices, Invertible Matrix Theorem (Nicholson Section 2.4)

Alternate Video Access via MyMedia | Video Duration: 47:08

Description: Halloween lecture: instructor was mugged and replaced by angel. Started with the definition of “invertible matrix”. Reviewed the super-useful theorem which the matrix inversion algorithm is based on.

2:00 --- simple example: when is a diagonal matrix invertible?

5:50 --- three 3x3 matrices --- are the invertible? Note that if the first matrix is A then the second matrix is 2A and the third matrix is A^T. From the example, we suspect that if t is nonzero then (t A)^{-1} = 1/t A^{-1} and if A is invertible then (A^T)^{-1} = (A^{-1})^T.

11:45 --- stated theorem to this effect. The proof of the theorem uses the super-important theorem: if you can find C so that AC = I then voila --- you’re done --- you’ve shown that A is invertible, you’ve found the inverse of A, and you get to write A^{-1} = C. (You don’t even get to write A^{-1} until you’ve shown A is invertible.)  Basically, if you can find a matrix that does the job that an inverse should do then you’ve found the inverse.  

28:00 --- did a classic exam question: if A is a square matrix such that A^2 – A = 2 I, find A^{-1}.

31:45 --- Inversion does not “play nice” with matrix addition.

38:40 --- The Invertible Matrix Theorem. Given a square matrix, presented 6 equivalent statements. If any one of them is true then all of them are true and it follows that the matrix is invertible. The point: some of those statements are pretty easy to check! Note: item 5 is that the columns being linearly independent is something that you haven’t learnt about yet --- ignore this item for the moment. Item 6 is that the Column Space of A is R^n --- you haven’t learnt about the Column Space of a matrix --- ignore this item for the moment.