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The Inverse of a Square Matrix - Problem 1

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.

Let's talk about how to find the inverse of a square matrix. I have matrix A equals 1 2, 3 4. I want to find A inverse. So I start by creating an augmented matrix 1 2, 3 4. I augment it by adding 1 0, 0 1, and this is the 2 by 2 identity matrix.

So I need to do row operations on this augmented matrix. What I need to accomplish here is to get the identity matrix on the left. Whatever is on the right will end up being my inverse of A. So the first thing I want to do to get the identity over here, is I'm going to try to get a zero in this position. The way I'm going to do that is multiply the top row by -3, and add that to the bottom row.

So this is the notation; -3 times row 1 plus row 2. When you write this notation, it's assumed that whatever row number you end with, that's the row that's going to get replaced. So the top row will remain unchanged, 1 2, 1 0. The bottom row -3 times 1 plus 3 is 0. -3 times 2 is -6, plus 4 is -2. -3 times 1 is -3 plus 0 is -3. -3 times 0 is 0, plus 1 is 1.

Next step. I need to get a 0 here, because the identity matrix has a 0 up here. So I'm just going to add this row to this row. So remember you want to end with row 1, because that's the one that's going to be changed. Row 2 plus row 1. Row 2 will remain unchanged. I can write that down now. So I add row 2 to row 1, and I get 0 plus 1 is 1. -2 plus 2 is 0. -3 plus 1, -2. 1 plus 0, 1.

Finally I just need to get a 1 here, and then this will be the identity matrix. I can do that by multiplying this bottom row by -½. So -½ times row 2. That's the notation for scalar multiplication by the bottom row. The top row remains unchanged. 1 0, -2 1. I get 0 times -½ is 0. -2 times -½ is 1 by design, and then -3 times -½, 3/2. 1 times -½, -½, and so I'm done. This is the signal you're done. You get the identity matrix on the left, and whatever is on the right, that's your inverse. So A inverse equals -2 1, 3/2 -½.

Now just to recap, these are the procedures for finding the inverse of a matrix A. You want to set up an augmented matrix with matrix A on the left, and the identity on the right. So if matrix A is a 3 by 3, you want the 3 by 3 identity on the right. You use a sequence of row operations until you can get the identity matrix on the left. Whatever you end up with on the right, that's the inverse. So that's A inverse.

One thing you have to remember when you're finding the inverse of a matrix is not every square matrix has an inverse. So if you're not able to get the identity on the left, that means that the matrix is not invertible.

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