The Inverse of a Square Matrix - Concept
We use the identity matrix to calculate a square matrix inverse. In order to be invertible, a matrix must be square, and by finding the square matrix inverse, we can find the solution of a system of linear equations. A square matrix inverse, when multiplied on the left or right by the original matrix gives us the identity matrix.
I want to talk about the inverse of the square matrix. Let's take a look at an example. I have a equals 5,-3 7,-4 and b equals -4,3 -7,5. First let's multiply a times b so I have -20+21 is 1, I have 15-15 is 0, I have -28+28 is 0 and I have 21-20 is 1. This is the identity matrix i so when I multiply a times b I get the identity matrix.
Let's try b times a. I get -20+21 1, I get 12-12 0, I get -35+35 which is 0 and I get 21-20 which is 1 again I get the identity matrix i. There are some special relationship between matrices a and b since a times b equals i and b times a equals i we say that a is an invertible matrix and that b is the inverse of a and this is how we write it b equals a inverse. Now this is really important because this is as close as matrices come to having a "reciprocal" you know like the reciprocal of 5 is one fifth and we use numbers like that a lot when we're solving equations you know if you're using if you're solving 5x=10 you could multiply both sides by one fifth and so similarly when we're working on matrix equations which we will be in a little bit, we want to have the idea of an inverse matrix so we can solve these matrix equations and so any time we can find a matrix that does this we call it the inverse of a.
Now you should remember not all square matrices are invertible and that's also like numbers. Not all numbers have a reciprocal this one doesn't so as as we'll see in the future, sometimes you won't be able to find the inverse of a square matrix.