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This video is about matrices, multiplying them, how it works, and how it works on a calculator.
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So first I'm just going to start off with how it works to multiply matrices.
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So here we go.
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Matrix A is 3 to negative 1, 5.
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It's a 2 by 2 matrix because there are 2 rows and 2 columns.
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The next matrix is matrix B. And that is also a 2 by 2 matrix.
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It has 2 rows and it also has 2 columns.
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You usually talk about values and matrices from left to right starting with the top.
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So for instance matrix A you would say the values are 3, 2, and negative 1 and 5.
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And matrix B negative 2, 1, 7, and negative 3.
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Alright let's go ahead and multiply these matrices.
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Let's do A times B. So I already have that written out here.
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B times B has matrix A first and it has matrix B second, which makes sense.
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It's A times B. And we will find out later in this section or in this lesson that order does matter for multiplying matrices.
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Okay, so here's how you multiply a matrix, 2 matrices.
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So let's take the first row of the first matrix. The first row of matrix A is 3, 2.
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Then you take the first column of the second matrix.
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The first column of matrix B is negative 2, 7.
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And you multiply these going from left to right on the first matrix and from top to bottom on the second.
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Usually I use my fingers. Obviously it's a little tough on the smart board.
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But so I'm going to take 3 times negative 2. I'm going to get negative 6.
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I'm going to add that 2. I'm going to move across that row to get to the 2.
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I'm going to move down the column to get to the 7. 2 times 7 is 14.
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So again that was 3 times negative 2 plus 2 times 7.
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And that's how I got negative 6 plus 14, which equals 8.
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So my first entry in my multiply in my product matrix is 8.
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Okay. I'm going to use that same row with all of the columns.
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So I already used the first column of matrix B. I'm now going to use the second column of matrix B with the same first row of A.
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And I'm going to do it in the same exact way I did the last one.
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I'm going to do 3 times 1.
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Then I'm going to add that 2. I'm going to move over on the row to get to the 2.
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I'm going to move down that column to get to negative 3.
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Okay. 3 times 1 is 3. 2 times negative 3 is negative 6.
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And when I add 3 plus negative 6, I get negative 3.
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So that value in my product matrix is negative 3.
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All right. Now I'm done with that row of the first matrix.
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I'm going to move to the next row, negative 1.5.
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I'm going to multiply this with the same 2 columns like I just did.
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So first I'm going to multiply it by the negative 2.7 column.
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So I'm going to take negative 1 times negative 2.
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I'm going to add that to 5 times 7.
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Moving across the row, moving down the column.
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Again, I would use fingers if I think it's just a lot easier to use your fingers.
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So you get negative 1 times negative 2 is 2. Plus 5 times 7 is 35.
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So 2 plus 35 is 37.
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Then you're going to use the same row with the second column.
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So I'm going to use negative 1.5 row of matrix A.
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I'm going to multiply that with the 1 negative 3 column of matrix B.
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So negative 1 times 1 plus, move across the row to the 5,
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and move down the column to negative 3.
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Negative 1 times 1 is negative 1.
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Plus 5 times negative 3 is negative 15.
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You add negative 1 and negative 15. You get negative 16.
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So your final matrix, the result of this multiplication,
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is all right it down just below.
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It's 8, negative 3,
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37, negative 16.
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All right, now we're going to try this on a calculator.
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I have a TI Inspire. Many of you might have a TI84 Plus.
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It works similarly.
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The only problem is on a TI84 Plus, it just takes a little longer to get all the matrices plugged into your calculator.
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So let me go back up to the top.
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And these are my two matrices again.
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So on my calculator, I'm going to do,
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okay, so I want to do menu, I want to add a calculator.
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On the 84 plus obviously this is where it should start just on your normal calculator screen.
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And I want a 2 by 2 matrix.
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For me, I'm going to hit Control and then the time sign and it's going to bring up some templates.
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And I want the 2 by 2 matrix in the second row here.
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And I'm just going to type in the values.
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So that was 3, 2, negative 1, 5,
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3, 2,
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negative 1, 5.
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And I'm going to multiply that by another 2 by 2 matrix.
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So go back into the templates and pick that same template.
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And this one was negative 2, 1, 7, negative 3.
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Negative 2, 1, 7, negative 3.
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And once I hit Enter, it'll give me the answer.
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8, 3, 37, negative 16.
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And this was the same that I had.
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So the calculator just makes it a little easier.
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On the 84 plus you'll have to go into matrix and then you'll have to add 2 matrices.
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And you'll have to type in the dimensions, which are 2 by 2 for both of them.
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And then type in all the entries.
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And then once you get matrix A and matrix B for instance, then you can multiply them.
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What I could have done is I could have said A.
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And then I could have said colon equals.
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And then I could have typed in my matrix.
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And that's 3, 2, negative 1, 5.
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And now that's stored into my variable A.
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And then I could have done B.
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Well, there's another way of storing also.
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I guess I'll do that for the second one.
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And this one was negative 2, 1, 7, negative 3.
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And to store this one, I'm going to just hit my store key, which is control and then the variable button.
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I'm going to store that into B.
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So again, both ways work to store something into a variable.
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And now I can just do A times B.
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And it'll give me the same answer I got up above, 8, negative 3, 37, negative 16.
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And that's sort of what the TI-84 plus does.
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It's something very similar to that.
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That's what your screen should look like.
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So again, just a little more convenient on the Inspire.
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You don't have to go into all the menus and enter them and then exit out of them and then enter back in.
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And it's just a little quicker and easier.
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OK, I'm actually going to show you one more example.
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Not that one.
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Well, here's another example.
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I'm actually going to do this one straight into the Inspire calculator.
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So we have matrix A and we have matrix B.
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Matrix A is a 4 by 2 matrix because it has 4 rows and it has 2 columns.
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Matrix B is a 2 by 3 because it has 2 rows and 3 columns.
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Now the thing to watch out for to see if you can actually multiply 2 matrices together in a certain order
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is to look at which order you're multiplying them.
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If I do A times B, A was a 4 by 2 matrix and B is a 2 by 3.
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If those middle numbers are the same, then you can multiply.
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Otherwise, you cannot.
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They're both 2 in this case, so we can do it.
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So I'm going to go ahead and store these.
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I'm going to actually rewrite my previous ones.
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So I'm going to do A colon equals.
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And I want a matrix template.
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And this time, I want to move over a few to this one that looks like a 3 by 3 matrix.
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Actually, it's as much as I want it to be.
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So this actually I want a 4 by 2 matrix.
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I want to hit OK.
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This was 1, 2, 4, 7, 3, 8, 1, 0.
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Yes.
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And then my second matrix, I'm going to do the same thing.
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B colon equals.
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And I want the same template.
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And this time, I want to be a 2 by 3 matrix.
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And this one was 4, 5, 7, 1, 3, negative 4.
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OK.
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So I have both matrices stored in my calculator.
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Now I want to show you how you can actually go about not just multiplying these together.
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We know that you would just do A times B.
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But I want to show you how, again, the matrix multiplication works just to drive it home a little bit more.
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I'm going to use a different matrix template this time to show you that I want to multiply the first row of A times the first column of B.
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So that's the row is 1, 2.
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I'm going to go into my matrix templates.
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And I want just a row, 1 by 2.
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So that's 1, 2.
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And I'm going to multiply 1, 2 times.
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Now I want a column matrix.
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It's just a column.
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It's a 2 by 1 column.
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I'm going to multiply it by 4, 1.
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1, 2 times 4, 1 is 6.
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So basically all I did in that step was I multiplied the first row of matrix A, which was 1, 2,
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by the first column of matrix B, which was 4, 1.
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So what it did is it said 1 times 4 is 4, plus 2 times 1 is 2.
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So 4 plus 2 is 6.
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And that's how it got the 6.
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So 6 would be my first entry in my answer.
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Okay, so I could go ahead and do that whole thing.
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I'm going to spare you.
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I'm just going to do A times B.
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And as you can see, the first entry is a 6.
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And there's the answer to A times B.
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All right.
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So again, kind of a lot easier on the inspire to enter matrices, just not as much of a hassle.
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Okay.
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But as I was saying, A times B is not always the same as B times A.
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And like I said about the matrices dimensions, the inside dimensions have to be the same.
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In other words, the columns of the first matrix have to be the same as the row,
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number of rows of the second matrix.
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Okay.
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And finally, the last thing I want to talk about is the identity matrix.
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The identity matrix can be any number of rows and columns as long as it's a square matrix,
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which means it has the same number of rows as it has columns.
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And there have to be ones along the diagonal.
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So this is a 2 by 2 version of an identity matrix.
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A 3 by 3 version, again, would look the same except there would be three rows and three columns,
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and there would be ones along the diagonal and zeros everywhere else.
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So all of these are identity matrices.
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And you might be able to guess what an identity matrix means.
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All it means is that if you multiply anything by the identity, for instance, here's matrix A,
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I'm multiplying it by the identity matrix.
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You can go ahead and check it for yourself, actually do the multiplication out,
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but the answer is going to be 5, 10, negative 2, negative 8.
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You're going to get exactly what you started with back,
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because the identity matrix doesn't change the answer.
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It's like doing 6 times 1.
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What do you get back when you multiply by 1?
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The original thing you multiplied, which was 6.
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So same exact idea with matrices.
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You're multiplying by it, and it doesn't change a thing to multiply by it.
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Okay.
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Let me just do that one on the inspire just to show you.
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I'm going to bring up a template. There is a 2x2 matrix template.
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Okay, I'm just going to make up some numbers.
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It doesn't really matter again, because I know I'm going to get this back anyway.
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I need to do a decimal, negative 4.5, and then 0.6.
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Okay, and I'm going to multiply this by the identity matrix.
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I'll go back to a 2x2 matrix, and again, to be able to multiply this,
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multiply anything by an identity matrix, the dimensions have to match up.
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So I know this has to be a 2x2 identity matrix.
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So 1, 0, 0, 1.
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We want 1s on the diagonal, and 0s everywhere else.
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And you can see I got the exact same matrix back that I multiplied by originally.
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All right, I hope you enjoyed learning a little bit about how to multiply matrices,
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about how it works, and how it works on a calculator.