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3x3 Determinants - Problem 2 1,747 views

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

Let’s calculate a couple more determinants. This time I’ve chosen two special ones. Notice that the determinant in apart a, is very similar to part b. Only the first two row’s are going to change I want to see with the effect of that is on the value of the determinant.

Anyway, let’s start with the first one. Remember what I said before, you can expand a determinant along any row or column. And it’s really good to expand along rows or columns with 0’s. So I’m going to expand this one along that row.

Second row, two 0’s are going to make this really easy. So expanding along the second row, I have to remember that I need to use these signs; minus plus minus. So starting with 2, so minus 2 times, and then the minor associate with 2 is going to be -7 3 1 6. And then of course, plus 0 minus 0, it's going to be plus 0 times some of other determinant, minus 0 times some other determinant doesn’t matter. Those are just going to zero out, so this will be a lot easier having it done along this row.

So -2 times -42 minus 3. That’s -45 times -2 is 90. Nice and easy. Now in this example I got my 0’s in the top row. So I’m going to expand along the top row. And that will give me and remember along the top row its plus minus plus.

You know another way to remember plus or minus, is to take the row and column number and add them up. Like this entry 2, is row 1 column 1 1 plus 1 is 2. Whenever you got an even result, use a plus sign, whenever you got an add result use a minus sign, that’s another way to figure out what the sign is going to be. In this case it’s going to be plus. Plus 2 times the determinant -7 3 1 6, minus 0 times some determinant. Write this guy times its minor, this guy times its minor, doesn’t matter what its minor is. Plus 0 times that determinant. Those are going to zero out and I’ll just get 2 times -42 minus 3. So that’s 2 times -45 or -90. So I got -90 for this answer and 90 for this one. That tells me that the result of switching two rows, has the effect of changing the sign of the determinant.