Inverting 3x3 matrices | Wild Linear Algebra A 8 | NJ Wildberger

This is the very important 8th lecture in this series on Linear Algebra by N J Wildberger. Here we solve the most fundamental problem in the subject in the 3x3 case---in such a way that extension to higher dimensions becomes almost obvious.

What is the fundamental problem? It is: How to invert a linear change of coordinates? Or in matrix terms: How to find the inverse of a matrix?

And the answer rests squarely on the wonderful function called the determinant. Be prepared for some algebra, but it is beautiful algebra!

CONTENT SUMMARY: pg 1: @00:08 How to invert the change in coordinates; 3x3 matrix; 2x2 review;
pg 2: @02:12 importance of the determinant; determinant relation to tri-vectors;
pg 3: @05:40 different ways of obtaining the determinant;
pg 4: @09:44 solving the 3x3 linear system;
pg 5: @14:28 solving the system continued;
pg 6: @16:46 3x3 inversion theorem derived;
pg 7: @17:56 notation to help remember the 3x3 inversion formula; definition of the minor of a matrix;
pg 8: @20:16 Definition of the adjoint of a matrix; relationship of the inverse, determinant and adjoint of a matrix;
pg 9: @22:47 examples; determination of the adjoint; determination of the inverse; matrix times its inverse; the identity matrix;
pg 10: @31:39 example;
pg 11: @34:52 3x3 matrix operations;
pg 12: @38:09 why the inverse law works; properties of a 3x3 matrix; an invertible matrix;
pg 13: @40:10 Proposition: If 2 matrices are invertible then so is their product, and the inverse of the product is equal to the product of their inverses (rearranged); proof;
pg 14: @42:18 exercises 8.(1:2) ;
pg 15: @43:41 exercises 8.(3:4) ; (THANKS to EmptySpaceEnterprise)

Video Chapters:
00:00 Introduction
2:12 importance of the determinant
5:40 different ways of obtaining the determinant
16:45 Theorem (3×3 inversion)
20:16 Definition of the adjoint of a matrix
34:51 3×3 matrix operations
38:09 why the Inverse law works
40:09 Proposition
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[haniamritdas]

I was recently comparing mathematics to a mountain of knowledge. While recognising that there is always a higher peak over the next ridge somewhere, it is encouraging to hear you this a peak in the difficulty of linear algebra at this level.

Seeing your more recent work on the (exp> polyseries extensions, and also the use of linear algebrs in generating binomials using scientific workplace in a calculus two lecture -- all of which are brilliant and exciting -- why is there no similar application (or, is there?) for generating the coefficients for the terms of a factorial polynomial ? I know of two methods for doing this using algebraic techniques, besides simply expanding the products. I am very eager to understand linear algebraic tools well enough to answer this question.

At any rate I love these questions, so central to our understanding and our ignorance. The exponential relations remind me of machine tools, because with a lathe and a mill, a machinist can make the parts of a lathe, and a mill, and many other things besides. In the same way that the factorial relations are used to construct themselves in the polynomial context. This is also a family of relations that can address the issues of contiuum and infinities by separating them from repeating algebraic expressions arising from basis factorization. Such a rich field. I love your work.

[darlove]

Professor, I absolutely love your presentations. Thank you very much for them. What more can I say? :)

[alpozen]

I am not sure if you have heard of it but you should really come visit the mathematics village in Turkey founded by the mathematician Ali Nesin. I would love to follow your lectures there. It is a wonderful atmosphere endowed with beautiful nature and filled by math lovers.