In this video, we're going to focus on finding the determinant of a 2x2 matrix and also of a 3x3 matrix. So let's start with the basics, the 2x2 matrix. Here, let's say this is A, C, B, and D. So it's going to be equal to A times D minus...
B times C. That's how you can find the determinant of a 2 by 2 matrix. So let's work on some examples. Let's say if this is 3, 5, negative 4, 7. So it's going to be 3 times 7 minus 5 times negative 4. 3 times 7 is 21. Negative 5 times negative 4 is positive 20. When you add them, this will give you 41. Now it's your turn.
For the sake of practice, try this one. Negative 7, 8. 4, negative 3. So it's going to be negative 7 times negative 3. And then minus 4 times 8. Negative 7 times negative 3 that's 21 Negative 4 times 8 is negative 32 21 minus 32 is negative 11 So that's how you can find the determinant of a 2 by 2 matrix Now what about a 3 by 3 matrix? So let's say this is A1, A2, A3, and then B1, B2, B3, C1, C2, C3. So let's go over the formula first.
The first thing we want to do is get rid of the first row and the first column. And notice what you have left over. B2, C2, B3, C3. So, you're going to use A1, and it's going to be reduced to a 2x2 matrix.
So it's going to be A1 and then B2, C2, B3, C3. So when you cross out row 1 and column 1, you'll be left with B2, C2, B3, C3. Now let's move on to the next one. So it's going to be minus. So instead of using A1, we're going to use B1.
B1 is in the first row and the second column. So what's left over is A2, A3 and C2, C3. That's going to be in the next 2x2 matrix. So it's going to be negative b1 and then a2, a3, c2, c3. And then plus.
The next one is going to be c1. c1 is in row 1, column 3. So what we have left over is a2, b2, a3, b3. So it's going to be c1.
a2 a3 b2 b3 and then you know how to evaluate a two by two matrix because we cover that already so now let's work on an example so let's say we have two four negative three five seven six negative 81 and nine So feel free to pause the video and work on this example. So we're going to use 2 first. So it's going to be...
We're going to put the 2 in front. And once we use the 2, we need to get rid of row 1 and column 1. So we're going to use 7, 6, 1, 9. So that's going to be inside the 2 by 2 matrix. And then minus.
Now we're going to use the 4. And we're going to get rid of row 1, column 2. So we're going to have 5, negative 8, and 6, 9. So it's a minus 4. And then 5, negative 8, 6, 9. Now the next number that we have is negative 3. So we're going to get rid of row 1 and column 3. So we have a negative 3 on the front. And then we're going to write what we see here. 5, 7, negative 8, 1. So that's how you can simplify the 3 by 3 matrix into 3 2 by 2 matrices. Now let's evaluate this 2x2 matrix. So it's 2, and then it's going to be 7 times 9 minus 1 times 6. So multiply these two first, 7 and 9, and then minus 1 times 6. The next one's going to be 5 times 9 minus negative 8 times 6. So first we have a negative 4, and then 5 times 9 minus 6 times negative 8. And then we're going to have negative 3. times 5 times 1 which is 5 minus negative 8 times 7 which is negative 56 so 2 times I mean 7 times 9 is 63 1 times 6 is 6 5 times 9 is 45 Negative 6 times negative 8, that's positive 48. And then we have 5 minus negative 56, which is like 5 plus 56, so that's 61. 63 minus 6, that's 57. 45 plus 48, that's going to be 93. Negative 3 times 61. is negative 183. 2 times 57 that's 114 and 4 times 93 that's 372 minus 183. So if we go ahead and combine these last three numbers this is going to give us negative 441. So that's going to be the determinant of the 3x3 matrix.
Let's go ahead and work on another example. Let's say this is 5, 7, negative 8, 4, negative 3, 6, 1, 7, and negative 9. Go ahead and calculate the determinant of this 3 by 3 matrix. So the first number is going to be 5. And if we get rid of the first row and the first column, we're going to get negative 3, 6, 7, negative 9. Next, we're going to use the 7. So, if we take away the first row and the second column, it's going to be 4, 1, and 6, negative 9. And then the last one, negative 8. So if we take away row 1, column 3, we're left with 4, negative 3, 1, 7. Now let's go ahead and evaluate the 2 by 2 matrices. So first we have negative 3 times negative 9, which is positive 27, and then minus 7 times 6, which is 42. And then it's going to be negative 7 times 4 times negative 9. That's negative 36. Minus 6 times 1, which is 6. And then this is going to be minus 8. 4 times 7 is 28. And then 1 times negative 3. It's going to be minus negative 3. 27 minus 42, that's negative 15. Negative 36 minus 6 is negative 42. And 28 minus negative 3, that's 28 plus 3, that's 31. 5 times negative 15 is negative 75. Negative 7 times negative 42, that's positive 294. and 8 times 31 that's negative 248 so if we combine these three numbers this is going to give us negative 29 as a final answer So that's the determinant of the 3x3 matrix. By the way, if you want to get more videos on algebra, trigonometry, precalculus, physics, general chemistry, organic chemistry, if you want to find more videos on these topics, feel free to check out my playlist.
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