Hey, everyone. Welcome back. So we've already seen how to calculate the determinant of a 2 by 2 matrix. And the way we did this was by multiplying the diagonals and subtracting. So we go down to the right and then down to the left. Now we're going to take a look at how to calculate the determinant of a 3 by 3 matrix, and I wish I could tell you that it's just as simple as just going down to the right and down to the left. But the bad news is it's a little bit more complicated. The good news, however, is that it actually does involve calculating a bunch of 2 by 2 determinants, which we do know how to solve. So I'm going to break it down for you and show you the equation for how to solve a 3 by 3 determinant, and then we'll take a look at an example together. Alright? Let's get started. So if I have a 3 by 3 matrix that's organized over here, I'm going to organize all the letters, you know, into subscripts, like a1 b1 a2 b2 c1 c2 c3, so on and so forth. Really, all it is to calculate a 3 by 3 determinant is you're going to have to calculate three numbers, so these three numbers. And I want to mention something here. The signs of these numbers will actually alternate. Notice how in this one, we have a plus sign, a minus sign, and a plus sign. So the signs will flip. Alright? And now what I want to do is sort of give you, like, an understanding of what this equation actually comes from. So if you have a 3 by 3 determinant, here is what's basically going on. Right? I'm going to write this same exact matrix out three times, and we'll see a pattern that starts happening with how these numbers where these numbers really come from or these terms. So for the first time, what you're going to do is you're going to take the a in the first row, and then you're going to strike out all of the other entries or all the other numbers that are in that column and row. So you're going to strike out the a column and the rest of the first row. And what you're left with is you're left with a smaller 2 by 2 matrix over here with the b and c numbers. This thing over here is basically what this becomes. You're going to take this a number, and you're going to multiply it by the smaller matrix that you've just come up with, and that's the first term. Let's take a look at the second one. The second one is you're going to take the b number in the first row, and again, you're going to strike out everything that's part of that column and that row. Strike out the rest of the b numbers and then the rest of the first row. So now we're going to take this number and multiply it by the smaller matrix that's left over, which is just made up of the a and c numbers. Right? So just draw a little box up over here. That's what the second number or the second number ends up being. It's b1 times the smaller matrix of the smaller determinants of the four numbers that you're left with. Now for the last one, you might have guessed, it's actually just going to be the c number in the first row. Now you strike out everything that's part of that row, and what you're left with is you're left with this matrix over here. So for the 3rd term, you're going to take that c and multiply it by the determinant of the smaller matrix that you're left with. That's what that third term ends up being. Alright? Now I know it's kinda complicated with all these letters and numbers, but that's just an easy sort of way to visualize what's going on here. And, hopefully, you'll sort of remember this pattern. You take the first numbers in each one of the rows, a, b, and c, and you strike out everything else that's part of that column and row and multiply it by the matrix, that smaller 2 by 2 matrix that you're left with. Let's go ahead and, actually, do this example real quick here. Alright? We can see how this works. So here, we got these numbers 3,1,0, -2, -3, -1, -4, 6. I want to calculate the determinants. Alright. So, the determinants of this matrix, remember, I'm going to take three numbers over here. The first one is going to be this 3. So, I'm going to take this 3, and then what I'm going to do is I'm going to strike out all the numbers over here and here. And what I'm left with is I'm left with this smaller matrix. So that's the determinant, so that's the smaller 2 by 2 matrix I'm going to multiply this thing by. So it's going to be (2, -3), (-4, 6). Alright? That's the first number. So now what I'm going to do is, remember, now the sign alternates, so I'm going to have to put a minus sign here. And now what's the next number? Well, the next number here is going to be the 1. Alright? So this is like the b number over here. Strike out everything in that column and row, and what I'm left with is this smaller determinant over here. So this is going to be 1, and then I'm going to have the smaller determinant, which is going to be (0, -3) and this is going to be (-1, 6). Alright? And now finally so that's the second number. Now finally, we just sort of undo all of that stuff. And now we're going to take a look at this number over here. So this is going to be remember, we're going to flip the sign. It's going to be plus. We're going to have 0, and then we're going to have the determinants of this matrix, after we cross out all of these other numbers. So this is going to be (2, -1), and (-1, 4). Now what's nice about this 3rd term here is that we're going to have 0 times the determinant of which is going to be some number. So 0 times anything is just going to be 0. Alright? So that third term actually just completely goes away, and that's good for us. This means there's less calculating. But that's really all there is to it. We just reduced this 3 by 3 determinant into a bunch of multiplications and 2 by 2 determinants. Now the rest, you already know how to do. We really just have to calculate these smaller 2 by 2 matrices over here, and we'll just go ahead and do that real quick. Alright? So what this is going to be is this is going to end up being we're going to expand this. This is going to be 3, and then remember we're going to have a bracket here because now we're going to need to evaluate these smaller determinants. So remember, you always go down to the right and then down to the left. So then in other words, this is going to be 2 times 6 minus (-3) times (-4). Alright? So just keep track of the minus signs. And that's what that first term will end up being. I'm going to have minus, and this is going to be 1 times, and then we're going to have bracket again. This is going to be 0 times 6. Right? So this is going to be 0 times 6 minus (-3) times (-1). Go down to the right, down to the left. Alright? So now let's simplify one step further. So here, what we're going to do is this is going to be equals. So I've got 3. And then over here, what I've got is I've got 12, so that's 2 times 6, minus and this is going to be -3 times -4, which is positive 12 as well. So in other words, 12 times 12. So what happens is this first term actually just goes away because this ends up just being 3. This ends up being 3 times 0, So that whole first term actually just goes away as well. So that whole thing just gets canceled out. And now what we're left with is we're left with 1, and this is going to be 0 times 6, which is just 0, minus, and then we have -3 times -1. So this is just going to be minus 3. Alright? So in other words, what this ends up being here is you end up just getting a grand total of 0 minus 1 times -3. Alright? So there's a lot of negatives here. So it's minus 1 times -3. So in other words, this ends up just being equals 0 +3, in which case the determinant of this whole 3 by 3 matrix, remember, it's just a number. It's just going to be the number 3. Alright? And the way we got here is we we was we reduced it to 3 smaller 2 by 2 matrices and calculated those. Alright. So that's really all there is to it, folks. Hopefully, this made sense. Thanks for watching, and I'll see you in the next one.
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Determinants and Cramer's Rule
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