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We've learned about matrix addition, matrix subtraction,

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matrix multiplication.

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So you might be wondering, is there the

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equivalent of matrix division?

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And before we get into that, let me introduce

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some concepts to you.

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And then we'll see that there is something that maybe isn't

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exactly division, but it's analogous to it.

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So before we introduce that, I'm going to introduce you to

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the concept of an identity matrix.

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So an identity matrix is a matrix.

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And I'll denote that by capital I.

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When I multiply it times another matrix-- actually I

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don't know if I should write that dot there-- but anyway,

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when I multiply times another matrix, I

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get that other matrix.

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Or when I multiply that matrix times the identity matrix, I

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get the matrix again.

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And it's important to realize when we're doing matrix

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multiplication, that direction matters.

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I've actually given you some information here that-- we

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can't just assume when we were doing regular multiplication

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that, a times b is always equal to b times a.

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It's important when we're doing matrix multiplication,

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to confirm that it matters what direction you do the

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multiplication in.

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But anyway, and this works both ways only if we're

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dealing with square matrices.

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It can work in one direction or another if this matrix is

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non-square, but it won't work in both.

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And you can think about that just in terms of how we

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learned matrix multiplication, why that happens.

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But anyway, I've defined this matrix.

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Now what does this matrix actually look like?

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It's actually pretty simple.

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If we have a 2x2 matrix, the identity matrix is 1, 0, 0, 1.

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If you want 3x3, it's 1, 0, 0, 0, 1, 0, 0, 0, 1.

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I think you see the pattern.

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If you want a 4x4, the identity matrix is 1, 0, 0, 0

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0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1.

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So you can see all that any matrix is, for a given

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dimension-- I mean we could extend this to an n by n

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matrix-- is you just have 1's along this top left to bottom

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right diagonals.

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And everything else is a 0.

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So I've told you that.

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Let's prove that it actually works.

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Let's take this matrix and multiply it

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times another matrix.

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And confirm that that matrix doesn't change.

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So if we take 1, 0, 0, 1.

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Let's multiply it times-- let's do a general matrix.

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Just so you see that this works for all numbers.

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a, b, c, d.

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So what does that equal?

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We're going to multiply this row times this column.

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1 times a plus 0 times c is a.

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And that row times this column.

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1 times b plus 0 times d.

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That's b.

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Then this row times this column.

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0 times a plus 1 times c is c.

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Then finally, this row times this column.

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0 times b plus 1 times d.

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Well, that's just d.

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There you have it.

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And it might be a fun exercise to try it the other

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way around as well.

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And actually it's an even better exercise to try this

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with a 3x3.

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And you'll see it all works out.

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And a good exercise for you is to think about why it works.

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And if you think about it, it's because you're getting

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your row information from here and your column

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information from here.

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And essentially, anytime you're multiplying, let's say

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this vector times this vector, you're multiplying the

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corresponding terms and then adding them, right?

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So if you have a 1 and a 0, the 0 is going to cancel out

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anything but the first term in this column vector.

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So that's why you're just left with a.

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And that's why it's going to cancel out everything but the

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first term in this column vector.

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And that's why you're left with just b.

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And similarly, this will cancel out everything but the

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second term.

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That's why you're left with just c there.

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This times this.

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You're just left with c.

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This times this.

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You're just left with d.

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And that same thing applies when you go to

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3x3 or n by n vectors.

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So that's interesting.

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You have the identity vector.

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Now if we wanted to complete our analogy-- so

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let's think about it.

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We know in regular mathematics, if I have 1 times

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a, I get a.

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And we also know that 1 over a times a-- this is just regular

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math, this has nothing to do with matrices-- is equal to 1.

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And you know, we call this the inverse of a.

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And that's also the same thing as dividing by the number a.

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So is there a matrix analogy?

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Let me switch colors, because I've used this green a little

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bit too much.

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Is there a matrix, where if I were to have the matrix a, and

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I multiply it by this matrix-- and I'll call that the inverse

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of a-- is there a matrix where I'm left with, not the number

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1, but I'm left with the 1 equivalent

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in the matrix world?

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Where I'm left with the identity matrix?

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And it would be extra nice if I could actually switch this

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multiplication around.

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So A times A inverse should also be equal to

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the identity matrix.

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And if you think about it, if both of these things are true,

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then actually not only is A inverse the inverse of A, but

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A is also the inverse of A inverse.

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So they're each other's inverses.

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That's all I meant to say.

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And it turns out there is such a matrix.

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It's called the inverse of A, as I've

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said three times already.

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And I will now show you how to calculate it.

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So let's do that.

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And we'll see calculating it for a 2x2 is fairly

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straightforward.

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Although you might think it's a little mysterious as to how

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people came up with the mechanics of it, or the

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algorithm for it.

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3x3 becomes a little hairy.

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4x4 will take you all day.

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5x5, you're almost definitely going to do a careless mistake

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if you did the inverse of a 5x5 matrix.

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And that's better left to a computer.

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But anyway, how do we calculate the matrix?

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So let's do that, and then we'll confirm that it really

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is the inverse.

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So if I have a matrix A, and that is a, b, c, d.

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And I want to calculate its inverse.

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Its inverse is actually-- and this is going

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to seem like voodoo.

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In future videos, I will give you a little bit more

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intuition for why this works, or I'll actually show you how

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this came about.

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But for now it's almost better just to memorize the steps,

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just so you have the confidence that you know that

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you can calculate an inverse.

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It's equal to 1 over this number times this. a times d

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minus b times c.

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ad minus bc.

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And this quantity down here, ad minus bc, that's called the

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determinant of the matrix A.

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And we're going to multiply that.

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This is just a number.

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This is just a scalar quantity.

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And we're going to multiply that by-- you switch

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the a and the d.

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You switch the top left and the bottom right.

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So you're left with d and a.

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And you make these two, you make the bottom left and the

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top right, you make them negative.

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So minus c minus b.

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And the determinant-- once again, this is something that

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you're just going to take a little bit on faith right now.

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In future videos, I promise to give you more tuition.

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But it's actually kind of sophisticated to learn what

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the determinant is.

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And if you're doing this in your high school class, you

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kind of just have to know how to calculate it.

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Although I don't like telling you that.

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So what is this?

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This is also call the determinant of A.

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So you might see on an exam, figure out the

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determinant of A.

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So let me just tell you that.

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And that's denoted by A in absolute value signs.

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And that's equal to ad minus bc.

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So another way of saying this, this could be 1 over the

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determinant.

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So you could write A inverse is equal to 1 over the

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determinant of A times d minus b minus c, a.

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Anyway you look at it.

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But let's apply this to a real problem, and you'll see that

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it's actually not so bad.

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So let's change letters, just so you know it doesn't always

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have to be an A.

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Let's say I have a matrix B.

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And the matrix B is 3-- I'm just going to pick random

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numbers-- minus 4, 2 minus 5.

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Let's calculate B inverse.

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So B inverse is going to be equal to 1 over the

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determinant of B.

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What's the determinant?

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It's 3 times minus 5 minus 2 times minus 4.

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So 3 times minus 5 is minus 15, minus 2 times minus 4.

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2 times minus 4 is minus 8.

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We're going to subtract that.

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So it's plus 8.

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And we're going to multiply that times what?

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Well, we switched these two terms. So it's minus 5 and 3.

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And we just make these two terms negative.

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Minus 2 and 4.

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4 was minus 4, so now it becomes 4.

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And let's see if we can simplify this a little bit.

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So B inverse is equal to minus 15 plus 8.

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That's minus 7.

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So this is minus 1/7.

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So the determinant of B-- we could write B's determinant--

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is equal to minus 7.

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So that's minus 1/7 times minus 5, 4, minus 2, 3.

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Which is equal to-- this is just a scalar, this is just a

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number, so we multiply it times each of the elements--

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so that is equal to minus, minus, plus.

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That's 5/7.

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5/7 minus 4/7.

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Let's see.

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Positive 2/7.

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And then minus 3/7.

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It's a little hairy.

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We ended up with fractions here and things.

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But let's confirm that this really is the inverse

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of the matrix B.

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Let's multiply them out.

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So before I do that I have to create some space.

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I don't even need this anymore.

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There you go.

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OK.

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So let's confirm that that times this, or this times

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that, is really equal to the identity matrix.

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So let's do that.

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So let me switch colors.

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So B inverse is 5/7, if I haven't made

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any careless mistakes.

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Minus 4/7.

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2/7.

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And minus 3/7.

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That's B inverse.

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And let me multiply that by B.

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3 minus 4.

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2 minus 5.

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And this is going to be the product matrix.

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I need some space to do my calculations.

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Let me switch colors.

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I'm going to take this row times this column.

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So 5/7 times 3 is what?

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15/7.

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Plus minus 4/7 times 2.

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So minus 4/7 times 2 is minus-- let me make sure

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that's right-- 5 times 3 is 15/7.

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Minus 4-- oh right, right-- 4 times 2, so minus 8/7.

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Now we're going to multiply this row times this column.

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00:12:07,909 --> 00:12:16,639
So 5 times minus 4 is minus 20/7.

265
00:12:16,639 --> 00:12:25,389
Plus minus 4/7 times minus 5.

266
00:12:25,389 --> 00:12:32,759
That is plus 20/7.

267
00:12:32,759 --> 00:12:35,710
My brain is starting to slow down, having to do matrix

268
00:12:35,710 --> 00:12:38,360
multiplications with fractions with negative numbers.

269
00:12:38,360 --> 00:12:40,950
But this is a good exercise for multiple

270
00:12:40,950 --> 00:12:41,730
parts of the brain.

271
00:12:41,730 --> 00:12:42,100
But anyway.

272
00:12:42,100 --> 00:12:43,350
So let's go down and do this term.

273
00:12:43,350 --> 00:12:48,800
So now we're going to multiply this row times this column.

274
00:12:48,799 --> 00:12:53,679
So 2/7 times 3 is 6/7.

275
00:12:53,679 --> 00:12:56,789
Plus minus 3/7 times 2.

276
00:12:56,789 --> 00:13:00,409
So that's minus 6/7.

277
00:13:00,409 --> 00:13:01,289
One term left.

278
00:13:01,289 --> 00:13:02,329
Home stretch.

279
00:13:02,330 --> 00:13:06,570
2/7 times minus 4 is minus 8/7.

280
00:13:06,570 --> 00:13:13,500


281
00:13:13,500 --> 00:13:17,730
Plus minus 3/7 times minus 5.

282
00:13:17,730 --> 00:13:25,300
So those negatives cancel out, and we're left with plus 15/7.

283
00:13:25,299 --> 00:13:28,370
And if we simplify, what do we get?

284
00:13:28,370 --> 00:13:31,500
15/7 minus 8/8 is 7/7.

285
00:13:31,500 --> 00:13:33,309
Well that's just 1.

286
00:13:33,309 --> 00:13:35,519
This is 0, clearly.

287
00:13:35,519 --> 00:13:36,259
This is 0.

288
00:13:36,259 --> 00:13:38,679
6/7 minus 6/7 is 0.

289
00:13:38,679 --> 00:13:42,429
And then minus 8/7 plus 15/7, that's 7/7.

290
00:13:42,429 --> 00:13:43,659
That's 1 again.

291
00:13:43,659 --> 00:13:44,730
And there you have it.

292
00:13:44,730 --> 00:13:47,149
We've actually managed to inverse this matrix.

293
00:13:47,149 --> 00:13:50,009
And it was actually harder to prove that it was the inverse

294
00:13:50,009 --> 00:13:52,980
by multiplying, just because we had to do all this fraction

295
00:13:52,980 --> 00:13:54,850
and negative number math.

296
00:13:54,850 --> 00:13:56,610
But hopefully that satisfies you.

297
00:13:56,610 --> 00:13:59,430
And you could try it the other way around to confirm that if

298
00:13:59,429 --> 00:14:01,009
you multiply it the other way, you'd also get

299
00:14:01,009 --> 00:14:02,049
the identity matrix.

300
00:14:02,049 --> 00:14:04,109
But anyway, that is how you calculate the

301
00:14:04,110 --> 00:14:05,600
inverse of a 2x2.

302
00:14:05,600 --> 00:14:08,950
And as we'll see in the next video, calculating by the

303
00:14:08,950 --> 00:14:12,310
inverse of a 3x3 matrix is even more fun.

304
00:14:12,309 --> 00:14:13,849
See you soon.

305
00:14:13,850 --> 00:14:14,000