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

Question

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

$∣ \begin{matrix} 3 & - 6 & 5 & - 1 \\ 0 & 2 & - 1 & 3 \\ - 6 & 4 & 2 & 0 \\ - 7 & 3 & 1 & 1 \end{matrix} ∣$

Accepted Answer

Hello, today we're going to be finding the determinant of the following four by four matrix. Now, the best way to take the determinant is to transform the matrix into an upper triangular matrix. What this means is that we're gonna turn the lower left hand corner into all elements of zero. And the reason for the spin is if we have an upper triangular matrix, then the determinant of the four by four is just going to be the product of the diagonal of the matrix. So how do we go about turning this four by four into an upper triangular matrix? Well, we already have one of the elements that we need to be zero. So let's go ahead and turn the element underneath that zero into zero. So we're going to need to change row three. And what we can do to make that element zero is to multiply the first row by a half, then add those values to row three. So we're gonna be changing row three and we're not changing the other three rows. So let's go ahead and write down the other three rows. Since we are not changing those values, we have four, negative 10 7, negative 208, negative five and three negative 122 and one. Now, if we multiply the elements of the first row by a half, then add those values to the elements in the third row. The matrix is going to change to 13/2 and negative one. Now this is convenient because another element that we need it to be zero has been changed to zero. So now what we're going to do is we're going to change this negative 1 to 0. So we need to change an element in row four. And the best way we can do that is multiply the elements in the first row by 1/4 then add those elements to row four. So the other three rows are going to be unchanged. Let's go ahead and copy down those values. Now, since we're doing this operation, this element is gonna change to zero and the sum of the remaining elements is going to be negative 1/2 15/ and 1/2. Now, what we wanna do is we want to change the element of negative one half to zero. So once again, we're changing another element in row 4 to 0. And what we can go ahead and do is multiply row two by 1/16 and add those values to row four. So we still have the first three rows unchanged. And since we're multiplying row two by 1/16 and adding those values to 14 to row four. The elements that we are left with is 55/16 and 11/16. Finally, we want to change this element of 55/16 to 0. So we're changing an element in row four again. And the best way we can do that to change it to zero is to multiply negative 55/100 and four to row three and add that to row four. So now let's go ahead and copy down the remaining elements. We have four negative 10, 7, negative 208, negative five and 13/2 negative one. And finally, we get 000 in row four. And the final element is going to be 253 over 208. So the lower left hand corner of this matrix has been changed to elements of zero. And now what we have is an upper triangular matrix. So that means that the determinant of this matrix is going to be the product of the diagonal of the matrix. So the determinant is going to be four times eight times 13/2 times 253/208. If we multiply these values together, we get the determinant of 253. And this is going to be the determinant of the four by four matrix. And with that being said, the answer to this problem is going to be c 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.

Use the determinant theorems to evaluate each determinant. See Example 4.
$∣ \begin{matrix} 3 & - 6 & 5 & - 1 \\ 0 & 2 & - 1 & 3 \\ - 6 & 4 & 2 & 0 \\ - 7 & 3 & 1 & 1 \end{matrix} ∣$

Key Concepts

Determinant of a Matrix

Determinant Theorems

Row Operations and Their Impact on Determinants

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