I'm wondering if the eigenvalue approach can invert a matrix:
T-1 A T = D
A = T D T-1
A T D-1 T-1 = I
Thus A-1 = T D-1 T-1
Does it?
Problem is that in doing that method is that you need to diagonalise the matrix A (i.e. find the diagonal eigenvalue matrix D, and the eigenvector matrix T). While finding D and T isn't hard, you will also need to find T^-1, so using that method to find an inverse for A requires you to find an inverse for T (which is of the same size as A), so that doesn't help.
anyway, just wondering if anybody learns methods to find the inverse of a matrix larger than 2x2? we learn how to invert a 2x2 using a formula but any bigger is 'beyond the scope of the course'. i know you can do it on Maple, but I'm curious about the non-computer method.
One way is Gaussian elimination, like in Affinity's post. That's a good method when your matrix has numbers in it (especially when the questions are rigged so that the answers turn out to be neat whole numbers), but not so good when there's pronumerals and variables in your matrix (doing Gaussian elimination on matrices with variables in it is a pain in the arse). I prefer the adjoint method in that case. The standard method for inverting a 2x2 matrix (swapping one diagonal, negating the other, then dividing by the determinant) is a special case of the adjoint method for 2x2 matrices. This website explains it:
http://home.earthlink.net/~jimlux/radio/math/matinv.htm
why is there such a big focus on matrices?
Depends on what you study, but matrices are one of the most important things to learn in maths for a lot of fields. For example in electrical engineering matrices are used heaps to represent & analyse circuits, about 80% of the maths involved is working with matrices in one form or another.
the other ways
