find the determinant of the matrix. Expand by cofactors using the indicated row or column. [

Find the determinant of the matrix. Expand by cofactors...

Key Concepts

Cofactor Expansion

Cofactor expansion is a method for calculating the determinant of a matrix by expanding it along a specific row or column. This technique involves multiplying each element of the row or column by its corresponding cofactor and then summing the results.

Minors and Cofactors

A minor of an element is the determinant of the submatrix formed by deleting the element's row and column. The cofactor is obtained by multiplying the minor by (-1) raised to the power of the sum of the row and column indices of the element, which helps account for sign changes during the expansion process.

Determinant

The determinant is a scalar value that can be computed from a square matrix. It gives important information about the matrix such as whether it is invertible and how much the linear transformation associated with the matrix scales area or volume.

Transcript

0:00 Hello there.

00:01 So here we got this matrix, 3x3 matrix, and we need to compute the determinant by using the expansion by cofactors.

00:10 So let's start by taking as reference the second row.

00:17 So the row in this case that we're going to take into account is two.

00:20 So this row, basically, we need to focus on this.

00:25 So just to remember how it's done the, how is computed the determinant of a matrix by expanding by expansion of cofactors is basically in this case we need to take a reference a row or a column in this case the second row so the formula is a to 1 c to 1 plus a 2 2 2 c 2 and plus a 2 3 c 2 3 you can notice that here we can change the if we change the row then this formal change.

01:07 We can either choose a row or a column and we're going to see that the result will not change.

01:15 So what is, so the entries, this a11, a2, a2, a2, a23 corresponds to the entries of the matrix.

01:26 And here the c21, c2 and c23 corresponds to the cofactors.

01:32 Just to remind you, the cofactors say ij is computed as minus 1 and the power of the summation of the indices and the minor.

01:44 Okay, so basically this formula for the co -factor is just taking the minor.

01:49 And in case that the summation of this indices is off, then we need to change the sign to the minor.

01:57 Otherwise, we just copy the value.

01:59 In case that i plus j is even, we just need to copy the value of the minor.

02:05 So that's how we're going to compute here.

02:08 This determinant.

02:12 So let's start with c2 .1.

02:15 Okay, sorry, but we need to also take into account what are the values for these entries.

02:22 So a to 1 corresponds to this entry on the matrix and is equals to 0.

02:28 So it doesn't make sense to compute the co -factor c21 because the result at the end will be 0.

02:34 So we just need to focus on c2 and c -23.

02:37 So c -2, the summation of these numbers, are is four so we don't need to change anything so we need to compute the minor so that means taking here this position to do eliminating the second row and the second column here and then a pyramid matrix the matrix 5 minus 3 1 and 3 and we need to take the determinant of this matrix and the determinant of this matrix is 18 and we need to repeat this procedure with the next co -factor.

03:19 But let me erase here these lines.

03:27 So 12, 0, and.....

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