This is the 8th lecture in this series on Linear Algebra. 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 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!