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 b a123, b123, c123, 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 these what this equation actually comes from. So if you have a 3 by 3 determinant, here's 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 a1 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 b1 number in the first row, and, again, you're going to strike out everything that's part of that column in 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 third term, you're going to take that c and multiply it by ``` 2 1 4 6 ```
Table of contents
- 0. Review of Algebra4h 16m
- 1. Equations & Inequalities3h 18m
- 2. Graphs of Equations43m
- 3. Functions2h 17m
- 4. Polynomial Functions1h 44m
- 5. Rational Functions1h 23m
- 6. Exponential & Logarithmic Functions2h 28m
- 7. Systems of Equations & Matrices4h 6m
- 8. Conic Sections2h 23m
- 9. Sequences, Series, & Induction1h 19m
- 10. Combinatorics & Probability1h 45m
7. Systems of Equations & Matrices
Determinants and Cramer's Rule
Video duration:
7mPlay a video:
Related Videos
Related Practice
Determinants and Cramer's Rule practice set
![](/channels/images/assetPage/ctaCharacter.png)