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 matrix. So we're given a four by four matrix. And the easiest way to take the determinant is to use the Cofactor expansion. So the Cofactor expansion focuses on one of the rows which in this case, we're gonna be using the top row and the pattern of the top row is going to follow plus minus plus and minus. No. What we're going to do is we're going to take the value of 66 is gonna be multiplied by positive one. And then we're going to multiply six by the resulting three by three matrix which in this case is gonna be all the elements in the lower, right. So what we end up getting is going to be six times the determinant of the three by three matrix 050705037. Then we're gonna take zero multiply it by negative one and multiply it to a determinant of a three by three matrix. But because we're multiplying a determinant by zero, that's just going to be zero. So we get negative zero for that value. And the same rule applies for the next term in the first row, the next term is zero. And so if we multiply that by a determinant of a three by three, it just becomes zero, then we're going to take 11 multiply it by negative one which is going to be negative 11. And we're gonna multiply it by the following three by three matrix which is going to be the elements of the lower left matrix. So we get negative 11 times the determinant of the three by three matrix negative 205870003. Now, there's one thing that we can do in order to get the determinant of the three by threes, we can go ahead and also use Cofactor expansion to get the determinants. So I'm gonna label this as determinant one and I'm gonna label this matrix as determinant two. Now we're going to perform Cofactor expansion on the first matrix, we have and this is going to follow the pattern plus minus plus. Now again, we're going to use the first row, any element that we have, that's zero in the first row is just going to produce a product of zero. So we have zero minus five times the resulting two by two matrix which is going to be and the next element is going to be zero. So we're going to be adding zero to the determinant. So what we're going to do is we're going to take negative five multiply, the determinant of the two by two that is going to give us negative five times seven times seven minus zero times five, which is zero seven times seven gives us 49. And so what we are left with is negative five times 49 and negative five times 49 is going to give us a value of negative 245. So the determinant of the first matrix is going to be negative 245. Now we're going to get the determinant of the second matrix by using the cofactor expansion. So the second matrix is negative 205870003. We can go ahead and we can swamp the first row with the last row and we can rewrite this matrix as 003870, negative 205. And the convenience of this by using the Cofactor expansion on the first row is that the first two elements are going to be zero. So the determinant of this three by three matrix is going to be three times the resulting two by two matrix which is going to be 87, negative two and zero. Now this is going to be three times the determinant of the two by two, which is going to be three times the quantity eight times zero, which is zero minus negative two times seven, negative two times seven is negative 14 times negative one gives us positive 14. So we're left with three times 14 and three times 14 is going to give us the value of 42. So that means that the determinant of the second matrix is going to be 42. And if we plug this back into the original Cofactor expansion, what we are left with is six times the quantity two negative 245 minus 11 times the quantity 42. Now negative 245 times six is going to give us negative 1470 and negative times 42 is going to give us the value of negative 462. And finally negative minus 462 is going to give us a determinant of negative 1932. This is going to be the determinant of the four by four matrix C. And with that being said, the answer to this problem is going to be B 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.

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