Evaluate each determinant by expanding by cofactors.

Key Concepts

Cofactor Expansion (Laplace's Expansion)

Cofactor expansion, also known as Laplace's expansion, is a technique used to compute the determinant of a matrix. This method involves selecting a row or column and summing the products of each element with its corresponding cofactor, thereby breaking down the problem into smaller, more manageable determinants.

Determinant

A determinant is a scalar value that can be computed from the elements of a square matrix. It provides important information about the matrix properties, such as invertibility and the scaling factor of the linear transformation represented by the matrix.

Minor

A minor is the determinant of a smaller matrix, formed by deleting one row and one column from the original matrix. It is used in the process of determining the full determinant of a matrix.

Cofactor

A cofactor is similar to a minor, but includes a sign factor, which is determined by the position of the element in the matrix using the formula (-1)^(i+j). This factor accounts for the alternating sign pattern needed in the expansion process.

Transcript

00:02 So here we need to find the determinant of this 4x4 square matrix by expansion of cofactors.

00:10 So expansion of cofactors tells us that the determinant will be a11, c11, plus a12, c12, plus a13, c14, c14.

00:29 So let's find each one of these elements.

00:31 So a11 is negative 1.

00:34 C -11, 101, 1 -0 -1, 3, negative -2 -5, negative -0 -0 -negative 1.

00:44 A -12 is 3 times the negative of the minor.

00:50 So 2 -0 -1, 1, negative 2, 5, 2 -0 -0 -0 -1, plus a -1, which is negative 2, 2, 2 ,000, times c -1, 2 ,000, 2 ,000, 2 ,000.

01:08 2 -1 -2 -13 -1 -15 -negative 1 plus a 1 -4 which is 5 and we delete this one so we have 2 -1 -2 -1 -3 -ne negative 1 0 negative 2 -0 oh but this would be the negative version of it sorry about that because that's the minor and a minor of c of minor m14 is the negative of the negative of c1 -4 is the negative of c1 4, so we have to convert that.

01:45 So now we have 3 by 3 matrices, so we'll have to convert them into 2 by 2 matrices.

01:51 So we have negative 1, and then i'm going to make brackets around all this.

01:59 So we have 1 times the minor, and then we have 0, so we don't have to write that because 0 times anything is automatically 0.

02:13 So we skip to a13, which is 1 times it's minus, so we have 3, negative 1, and then.

02:20 Negative 2, 0, and that's it.

02:24 So we put brackets around.

02:26 Now we have minus 3.

02:27 I'm going to bring that negative sign out.

02:32 A11 is 2, c11, negative 2, 05, negative 1.

02:41 Again, a12 is 0, so we move on to a13, which is 1, and delete this row.

02:48 So we have c13, which is 1, 2, negative 2.

02:55 Now we have negative 2, and we have 2.

03:01 As a11, c11, 3, negative 1, 5, negative 1.

03:10 Now a12 is 1, and then the negative side.

03:16 Why don't we just make this into a negative sign? so negative of minor 1, 2.

03:26 So we have 1, 2, 5, negative 1, and then a3, and c1 3.

03:39 And lastly, we have negative 5 over here.

03:45 And 2 times a11.

03:54 And then we have minus 1 because i'm going to bring that negative out from the minor.

04:02 So we have minus 1, 2, negative, 2, 0.

04:10 Now i'm going to distribute within the brackets.

04:14 So here we have negative, negative 2, 05, negative 1, minus 3, negative 1, negative 1, negative 1, negative 1, 2, 0.

04:28 Here we have minus 6, negative 2, 0, 5, negative 1...

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