One of the major uses of inverses is to solve a system of linear equations. You can write a system in matrix form as AX = B.
You must place the inverse of the matrix adjacent to the matrix. That is because Inverses need to be next to each other (very loose mathematically, but think back to functions) in order to undo each other.Now, pre-multiply both sides by the inverse of A. Make sure you meet these two conditions.
If you multiply by putting something in front of the left side (pre-multiply), it has to go in front of the right side. If you put something behind (post-multiply) the left side, it has to go behind the right side.
Matrix Multiplication is NOT Commutative!
A-1(AX) = A-1(B) ... pre-multiply both sides by A-1
(A-1A) X = A-1 B ... use the associative property to regroup factors
I X = A-1 B ... when you multiply inverses together, they become the identity matrix
X = A-1 B ... the identity matrix is like multiplying by 1.
If AX = B, then X = A-1 B
If XA = B, then X = B A-1
If AXC = B, then X = A-1 B C-1
So what you're asking in your normal cynical way is "You've just solved another equation, what does that have to do with anything?"