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The Inverse of a Square Matrix - Concept

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

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.

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