Evaluate the determinant of the matrix. [ -5 4 9 1 0 -2 0 -5

Evaluate the determinant of the matrix. [ -5 4 9...

Key Concepts

Minors and Cofactors

A minor is the determinant of the submatrix formed by deleting one row and one column from the original matrix, while a cofactor adjusts the minor with a sign that depends on the position of the element. These concepts are crucial for the cofactor expansion method and for understanding the structure and properties of a matrix.

Matrix Determinant

The determinant is a scalar value computed from a square matrix that encapsulates important information about the matrix, such as whether it is invertible, and describes how the matrix scales volumes and transforms space. It is a central concept in linear algebra with applications in solving systems of equations, understanding eigenvalues, and more.

Cofactor Expansion

Cofactor expansion is a systematic method for computing the determinant of a matrix by expanding along a row or column. This method involves calculating minors for each element in the chosen row or column and then combining them with a sign factor to produce the overall determinant.

Transcript

00:01 This question gives us a 3x3 matrix and asks us to find the determinant.

00:06 The matrix that it gives, which we'll call a, looks like this.

00:10 Negative 5, 49, 1, 0, negative 2, and 0, negative 5, 7.

00:19 So as always, when we find the determinant of 3x3 matrices, i'm going to choose a row or column and find the corresponding cofactors and scale them by the elements in that row or column, and then add them all together.

00:32 I'll write it out so you'll see what i mean more clearly.

00:34 I'm going to choose this row here, row number two, because it has a zero, which we know we like, because it will cancel out co -factors, and it has other small numbers that are easy to work with.

00:48 So the determinant of a is going to look like this.

00:52 It's going to be element 2 -1, which we know is 1, times co -factor 2 -1, plus element 2 -2, which of course is 0 times co -factor 2 -2, plus element 2 3, which is negative 2, times co -factor 2 3.

01:10 Now, right away, we know that this term cancels out because 0 times anything is just 0.

01:16 So we're left with only two co -factors that we need to find.

01:19 I'm going to find each one individually, then we'll add them all together.

01:23 So the first, i'm finding 1 times co -factor 2 -1, which is just equal to the co -factor 2 -1.

01:30 And we know co -factor 2 -1 is the same thing as negative 1 to the 2 plus 1.

01:37 Times the determinant of minor to 1.

01:41 Now what's minor to 1? remember, minors are, if we take the entire matrix and then omit the column and row that are specified.

01:49 So here, minor 21, we're going to omit column 2 and row 1.

01:55 And we're left with 4, 9, negative 5, 7.

01:58 So let me write that in.

01:59 We'll have negative 1 to the 3rd times the determinant of 4, 9, negative 5, and 7.

02:09 Of course, negative 1 to the 3rd, negative 1 to any odd number is going to be negative 1.

02:13 So we'll have negative 1.

02:14 And when we find the determinant of 2 by 2 matrices, but now you're a pro at this, remember we take the diagonal, the product of the diagonal, and then subtract from that the product of the antidiagonal.

02:24 So we'll have 4 times 7, which of course is 28, and then subtract negative 5 times 9, which is negative 45...

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