Identity and Inverse Matrices
At this point you may be wondering why the fourth of the Four Horsemen of the Math Apocalypse, division, seems to be getting overlooked. Well, the fact is that you cannot do division with matrices. That's okay, though, because if you can't divide by x, for example, you can multiply by its inverse:
The same is true with matrices. All we need to know is how to find the inverse, right?
Check this out in the context of solving an equation:
2x = 4
Normally we would just divide each side by 2 in order to solve this:
2x = 4, , and x = 2
But what if we can't divide? Think of it this way. Dividing by 2 in this example is the same as multiplying by the inverse of 2:
is the same as because is the inverse of 2.
Let's think this through with all variables.
If this is our equation:
ab = c
and we need to solve for b without division, we need to find the inverse of a and multiply both sides by that:
No division in sight. Ready to do this with matrices?
Finding the inverse is the first step. The easiest way we know to do it is to create this weird double matrix thing and go from there. Get THIS.
Let's say this is our matrix:
and we want to find its inverse. We need to look at it alongside an identity matrix to find that inverse:
Basically, we want to transform our target matrix into an identity matrix by multiplying whatever you need to across both. Then, those calculations you make end up turning the identity matrix you start with into the target matrix's inverse. Weird, but true.