Evaluate the determinant of the given matrix by first using elementary row operations to reduce

Evaluate the determinant of the given matrix by first using...

Key Concepts

Determinant

The determinant is a scalar value that can be calculated from the elements of a square matrix. It provides important information about the matrix, including whether the matrix is invertible and the scaling factor of the linear transformation represented by the matrix. For example, if the determinant is zero, the matrix does not have an inverse.

Elementary Row Operations

Elementary row operations include row swapping, multiplying a row by a nonzero scalar, and adding a multiple of one row to another row. These operations are used to simplify a matrix, often to reach an upper triangular form or reduced row echelon form, and they have specific effects on the determinant that must be accounted for.

Upper Triangular Form

A matrix is in upper triangular form when all the entries below the main diagonal are zero. Evaluating the determinant of an upper triangular matrix is straightforward because it is simply the product of the diagonal entries. Reducing a matrix to this form via row operations simplifies the calculation of the determinant.

Impact of Row Operations on Determinant

When using elementary row operations to simplify a matrix, it is crucial to understand how these operations affect the determinant. For instance, swapping two rows changes the sign of the determinant, multiplying a row by a scalar multiplies the determinant by that scalar, and adding a multiple of one row to another does not change the determinant. Being aware of these impacts helps in accurately computing the determinant after transformation.

Transcript

00:02 Section 3 .2, problem number 12, we're dealing with properties of determinants.

00:07 We've got a 4 -5 -4 matrix.

00:09 They've asked us to find determinants by using elementary row operations to reduce it to upper triangular form.

00:16 So i'd just like to go ahead and say, okay, everything in column 1 below that 7 needs to be cleared out to be a 0.

00:24 So the way to make this happen is negative 2 times row 1, add that to row 2.

00:33 And let that become the new row 2.

00:36 So when we do that operation, we have 7, negative 1, 3, 4.

00:45 So negative 2 times 7, negative 14, add to get 0.

00:50 Negative 2 times negative 1 is 2.

00:54 Add that to 2 and get 4.

01:00 Negative 2 times 3 is negative 6, add to 4, and you get negative 2.

01:06 Negative 2 times 4 is negative 8, add to 6, and get negative 2.

01:09 2.

01:11 Now while i'm here, the next thing i want to clear will be that 21.

01:15 So negative 3 row 1 plus row 3.

01:21 That will become my new row 3.

01:24 So negative 3 times 7 is negative 21.

01:28 I had to get 0.

01:29 Negative 3 times negative 1 is 3.

01:32 3 plus 1 is 4.

01:35 Negative 3 times 3 is negative 9 negative 9 plus 3 is negative 6 negative 3 times 4 is negative 12 negative 12 positive 4 is negative 8 and then the last thing i want to clear would be this negative 7 you can do that by adding row 1 to row 4 and letting that become the new row 4 so 7 plus negative 7 negative 4 plus 1 5 plus 1 5 plus 3 and 8 plus 4.

02:15 So now i've cleared everything in that first column beneath the diagonal, and now i just simply need to go forward and to clear everything below the 4, that first 4, in the second diagonal.

02:27 So let's just go ahead and break this into steps.

02:31 So i want to clear this 4 right here.

02:33 So that operation is going to be negative row 2 plus row 3 and let that become the new row 3...

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