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Simplifying Determinants - Problem 1

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.

I want to compute a determinant and I’m going to try to use some of my rules for simplification to make this as simple as possible. Remember that when we expand a determinant, we like to expand along a row or column that has some zeros in. This doesn’t have any zeros. So let’s creates some by doing some row operations.

One thing that we can do is, we can add a multiple of one row to another, and we can add a multiple of one column to another. What I want to do is I want to try and get a zero right here, by adding -2 times this last column to this column. So, that’s going to change the middle column but it won’t change the first or last. So I can write those down as they are. 2 3 6, and –2 times this is -4, plus 4 is 0. -2 times 3 is 6,, plus 9 is 3. -2 times 6 is -12, plus 8 is -4. So this column is -2 times C3 plus C2.

Let’s see if I can do that again, because if I have two zeros in one row, it’s going to be really easy to evaluate the determinant. Let’s try to get a zero right here. I’ll get that by multiplying this column by -4, and adding it to the first column. The second and third columns will stay the same; 0 3 -4, 2 3 6. And again I’m adding -4 times this column to this one. So -4 times 2 is -8 plus 8, 0. -4 times 3 is -12, plus 3, -9. -4 times 6 is -24, minus 2 is -26.

I got what I wanted. I have two zeros here. It’s going to be really easy to expand along this top row, when I finally do that. Let me also write down this was -4 times C3 plus C1. -4 times the third column plus the first column.

Now before I actually do this, evaluate this determinant, let’s factor out as many constants as we can. For example there’s a common factor of 3 in this row, and there’s a common factor of 2 in this row. Actually I can factor the two out of the top row. Let’s factor all those things out.

So pulling a 2 out of the top row, leaves 0 0 1. Pulling a 3 out of the middle row leaves -3 1 1. Pulling a 2 out of the last row give me -13 -2 3. That’s a nice simple determinant. I’ve got 12 times, I’ll put some parenthesis here. So I’m going to expand along the top row. It’s going to be plus 0 times this minor, doesn’t matter it’s zero. So plus 0 minus 0 times this minor. Doesn’t matter if zero. Plus 1 times its minor which is this determinant here. -3 1, -13 -2. I still have the 12 out in front.

This is going to be 6 minus -13, so 6 plus 13, is 19. It’s 12 times 19. 12 times 20 would be 240, so 12 times 19 is just 12 less, 228. That’s our answer.

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