If is a 2-by-2 matrix with then the matrix satisfies . We say that is the inverse of .
In the next few videos, we're going to answer the question: can you divide by a matrix?
This is the analogue of the reciprocal of a number: the equation is the analogue of .
Let's just check (check the other equality for yourself).
We can use this to "divide" by a matrix: if we have a matrix equation then we can multiply both sides on the left by to get , and since this means .
With great power comes great responsibility: you should never write for matrices ! It's not clear if you're doing or (and these are different because and might not commute).
We can use inverses to solve simultaneous equations. For example: is equivalent to where and . We can compute , so and this coincides with the solution , we found earlier.
We'd like to generalise the notion of inverse to bigger matrices.
Let be an -by- matrix. We say that is invertible if there exists a matrix such that . (Here is the -by- identity matrix). If such a exists, then it's unique, so we're justified in calling it the inverse of and writing it as .
To see that the inverse is unique when it exists, suppose you had two inverses for : Now evaluate in two ways:
If are invertible matrices then is invertible with inverse .
we have and so is an inverse for .
Had we tried to use instead, we would have obtained , and we couldn't have cancelled anything because the various terms don't commute.
We saw for 2-by-2 matrices that a matrix is invertible if . Even more basically, a 1-by-1 matrix is invertible if and only if . We'll see in a few videos' time that there is an analogous quantity (the determinant) which is nonzero precisely when is invertible. First though, we're going to explain how to calculate inverses of -by- matrices.