3x3 Determinants - Problem 1
Let’s calculate another determinant. So here is an example; Determinant of 2 1 -1 -1 2 1 0 4 2. Now in a previous example, we expanded the determinant along the top row. I just want to show you that you can actually expand along any row or column. You just have to follow some simple procedures. I want to expand along this column, so the way that works is one by one I take each of these entries and I calculate it’s minor. So for example the minor for 2 would be 2 1 4 2 1 4 2 that determinant that 2 by 2.
Then I look at the next one -1, I calculate its minor, it's 1 -1 4 2. And then the last one is 0 times, well it doesn’t really matter but I’ll write it out just for practice. 1 -1 2 1. And this is actually one of the reasons that you want to build or expand along any row or a column. Whenever you get a 0, you know that you are going to save yourself some work.
0 times whatever this is it's going to be 0. You got to use the alternating sign, the thing that we had to use in the previous example. Now here is how to remember, when you are dealing with a three by three determinant, and you are expanding along the first column. The first term has to be plus, the second one has to be minus and the third cone has to be plus. So plus, minus, plus.
And in general you can recreate this diagram yourself just by remembering that the upper left is always plus. And then you got to make checker board out of it. You have to have alternating signs both going across and down. So plus minus plus, plus minus plus like that so that you can actually do this for a four by four, or a five by five determinant if your teacher is that mean. Anyway you could always create this diagram to tell you how or to figure out the signs here. Now let’s actually compute this.
2 times 4 minus 4. 2 times 0 we’ll get that in a second. Plus 1 times 2 minus -4, that’s 2 plus 4, and of course that’s just plus 0. So that’s the beauty of expanding down a row or column with a 0. 2 times 0 is 0 also 6 plus 0 that’s 6 so the entire determinant is 6.