Matrix Inverse

Given M = AB, the inverse of M, inv(M), is: inv(M) = inv(AB) = inv(B)*inv(A).

For details of how you would derive this, see bottom of this page: http://mathworld.wolfram.com/MatrixInverse.html.

If M is invertible, and M = A*B, then:

Inverse(M) = Inverse(A*B) = Inverse(B)*Inverse(A)

Quote:

You can take the inverse by computing the inverse of each component, right?

If you mean what I think you're saying, then no. To find the inverse of a matrix A, you must find a matrix A' that when multiplied by A results in I, the identity matrix. This isn't typically found by inverting each component of a matrix.

Matrices are just functions, so when you see two matrices being multiplied together, say AB, you should read A of B. In other words, A(B) as a composition of functions. So B happens first, then A is applied to it.

AB = A(B) = M
A^-1(A(B)) = B = A^-1(M)
B^-1(A^-1(A(B)) = B^-1(B) = I = B^-1(A^-1(M)) => B^-1 A^-1 = M^-1

Which has already been stated here, but at least it's a decent explaination, and might help you with further matrix problems. :)

EDIT: This is assuming both A and B are in fact invertible. Not every matrix is, so be careful.

EDIT: Also, there are instances when B^-1 A^-1 = A^-1 B^-1. In fact, this is billboarding techniques were figured out. The matrices created by rotations and translations in the 4D vector space were...okay guys, help me out here. Homogenous?

Thanks for the help guys.