Find the value of each determinant.

Key Concepts

Cofactor Expansion

Cofactor expansion, also known as Laplace expansion, is a method for evaluating the determinant of a square matrix. It involves selecting a row or column and summing the products of each element by its corresponding cofactor, which includes the determinant of a smaller submatrix along with a sign determined by the element’s position.

Determinant

A determinant is a scalar value computed from a square matrix that captures key properties of the matrix, such as its invertibility, the volume scaling factor in linear transformations, and the uniqueness of solutions in linear systems. It plays a central role in linear algebra and is used to characterize the behavior of the corresponding linear map.

Transcript

00:01 So to find the determinant of this 3x3 -by -3 matrix, you're going to find the minors of all the elements here in the second column.

00:08 So the first element here in the second column is 3.

00:11 So we're going to find the minor of three.

00:14 So the minor, so the minor, so m sub 1, 2, right? the minor of the element in the first row, second column, is equal to the determinant of crossing out the first row and second column.

00:27 And looking at the remaining four elements.

00:28 So you get the determinant here of 2 -0 -negative 6 -2.

00:36 Okay, well, this is equal to, well, 2 times 2 minus negative 6 times 0.

00:51 So that is 4 minus 0, which is equal to 4.

00:55 So the first minor here, m sub 1 -2, is equal to 4.

00:59 Then we find our, well, similarly find the other two minors, so m sub 2 -2, is going to be equal to the determinant of the 2x2 matrix 3, negative 1, negative 6, and 2.

01:17 Okay, that's equal to, well, 3 times 2, so 3 times 2 minus minus 1 times minus 6.

01:28 That's equal to 6 minus 6, right? because this is positive, so that's 6 minus 6.

01:37 So this determinant or the minor here is equal to 0.

01:41 And then looking at our last minor, we have m sub 3, 2.

01:47 Okay, so we cross out our third row and second column.

01:50 Look at the remaining elements, and we have the determinant of the matrix 3, negative 1, 2, and 0.

01:59 So this determinant is going to be equal to 3 times 0 minus 2 times a negative 1.

02:09 Okay, so this is equal to, well, 3 times 0 minus a negative 2.

02:14 That's 3 plus 2, which is equal to 0...

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