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3x3 Determinants - Problem 3
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Let’s take a look at another example. This time I have two closely related determinants. I want you to see how they are related. First of all, they both have the exact same third row. And if you look at the first two rows of each. you’ll notice that in part b. the first row is 2 times the first row of part a. And the second row is 5 times the second row of part a. I want to see what effect, multiplying a row by a constant has on the result of the determinant. So I’m going to calculate both of these and see what that does.

First, let’s calculate part a. And because I got a 0 in the bottom row, I’m going to expand along the bottom row. So let’s remember what the whether I’m going to need pluses or minuses here. In the bottom row, this first entry is row 3 column 1, 3 plus 1 is 4. And whenever you get an even result for the row plus column, you use a plus.

So this is going to be plus 0 times some minor; 2 -2 1 6. And then I’m going to have minus 2, remember that the pluses and minuses okay. So minus 2 times the determinant 1 -2 3 6. Plus 5 times its minor and its minor is 1 2 3 1. So let me calculate that.

I’m going to get 0 here. And then minus 2 times 6 minus -6, 6 plus 6 is 12 plus 5 times. We have 1 minus 6 -5 and so this is going to be -24 minus 25 minus 49. So that’s the determinant of this matrix.

Now let’s compare this one. I’ll expand this one across the bottom row as well. So I’m going to get again plus 0 times some determinant, it doesn’t matter, minus 2 times, 2 -4 15 30. Plus 5, times 2 4 15 5. Let’s see what happens here.

I have 0 minus 2 times 60, minus -60. 60 plus 60 is 120. Then I have plus 5 times 10 minus 60, -50.So here I have minus 240, and here I have minus 250. That’s minus 490. So this answer is exactly 10 times this answer. Now why would that be? Look at the rows.

This row is 5 times that row. This row is 2 times that row. The product is 5 times 2 times this and that’s exactly what you can do when you are calculating the determinants. You can actually factor these numbers out. So this factor of 5 can be pulled out. Pulled out of the determinant. And this factor of 2 can also be pulled out. And that tells you that this determinant is exactly 10 times the other.

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