Use the determinant theorems to evaluate each determinant. See Ex...

Question

Use the determinant theorems to evaluate each determinant. See Example 4.

Accepted Answer

Hello, today we're gonna be finding the determinant of the following four by four matrix. So in order to get the determinant, there's a bit of simplification that we can do, we can simplify the first row by multiplying the fourth row by negative one, then adding those values to the first row. So this is going to change the four by four matrix. Now, since we're not changing the 2nd 3rd or fourth row, we're gonna go ahead and write down those rows right now. So we have 10 5, negative nine and 238, 11 and 17 65, negative four and eight. Now we're changing the values in row one by multiplying the fourth row by negative one, then adding those values to the first row. So six times negative one is going to give us negative six and negative four minus six is going to give us negative 10, five times negative one is going to give us negative five and zero minus five is going to give us negative five. Next negative four times negative one is going to give us positive four and five plus four is going to give us nine. Finally, negative eight, I'm sorry, positive eight times negative one is going to give us negative eight and six minus eight is going to give us negative two. Now notice that the elements in the 1st and 2nd row are very similar to each other. In this case here, the first row is different by the second row by a multiple of negative one. So we can further change the first row by multiplying the first row by negative one. And if we multiply negative one to the first row, we get 10, 59 and two, I'm sorry, positive 10, positive five, negative nine and positive two. And the second row, we have the exact same elements, then we have the third row and the fourth row. Now one thing to note about the determinant theorems is that if you have any repeating rows inside of your matrix, then the determinant is automatically going to be zero here. The first row and the second row are completely identical to each other. So that means by the determinant theorems, the determinant of the four by four matrix is going to be zero. And with that being said, the answer to this problem is going to be a. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

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Use the determinant theorems to evaluate each determinant. See Example 4.