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 three by three matrix. So what we are given is the three by three matrix with 02, negative 59731 negative six and 15. Now, one thing to note is that we want to use the determinant theorems. And one way to simplify this three by three matrix is to go ahead and change row three, we can simplify row three by multiplying row one by row three and adding those values to row three, this is going to change the matrix or at least the third row of the matrix. So we're gonna write down the first two rows. So since those are staying unchanged, so we're gonna have 02, negative 5973. But since we're changing the fir the third row three times zero is going to give us zero and zero plus one is going to give us one three times two, is going to give us six and six minus six is going to give us zero and negative five times three is going to give us negative 15. A negative 15 plus 15 is going to give us zero. So after doing this step, we, we are able to simplify our third row to be 100. Now, the next thing into simplifying this three by three matrix is to swap the positions of row one and row two doing this is going to give us a new the by three matrix which is going to be 02 negative five. And now we're going to use the method of Cofactor expansion to get the determinant of the three by three matrix. So using Cofactor expansion, we're going to follow the pattern of plus minus plus for the first row of the elements. If we use Cofactor expansion, we're going to take the first element which is one and multiplied by the two by two matrix of the lower four elements that is going to be multiplied by the two by two determinant of 732 and negative five. And now we're gonna repeat this process for the next element. But notice that for the Cofactor expansion, we're gonna be multiplying zero by the determinant of and three negative five, it doesn't matter what this two by two determinant is going to be since it's being multiplied by zero, it's just going to cancel out to give a zero. So we're going to be subtracting the value of zero to the Cofactor expansion. And the same thing applies to the last element. If we take the next element zero and multiply it by the determinant of the lower left elements. It doesn't matter what this determinant is going to be. Since it's multiplying by zero, it's going to simplify to give us zero. So what we are left with is one times the determinant of the two by two matrix 732 and negative five. In order to take the determinant of a two by two matrix, we're gonna multiply the lower, upper left and lower right element and subtract the product of the lower left and upper right element. That is going to give us one times the group seven times negative five minus two times three. Now seven times negative five is going to give us negative 35 and negative two times three is going to give us negative six. Finally negative 35 minus six is going to give us negative 41 and negative 41 times one is just going to give us negative 41. So the determinant of this three by three matrix is going to be negative 41. 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.