CHAPTER 04.06: GAUSSIAN ELIMINATION: Determinant of a Matrix Using Forward Elimination Method: Background
In this segment, we're going to talk about how to find the determinant of a matrix. There are several ways to find the determinant of a matrix, but one of the most computationally efficient ways of finding the determinant of a matrix is by using the same steps of forward elimination as in doing Gauss elimination. So letís suppose you want to find out the determinant of an n-by-n matrix. What you're going to do is you're going to apply the steps of forward elimination of Naive Gaussian method, so you're going to apply the forward elimination steps, and what you're going to end up with is an upper triangular matrix. So if you look at the forward elimination steps of Gauss elimination, which is explained in another segment, I'll be annotating it to that particular segment.
So you take a coefficient matrix, n-by-n, and you end up with an upper triangular matrix, n-by-n. Now, what happens is that when you are conducting forward elimination steps on the coefficient matrix, the only thing which you are doing is that you are adding or subtracting a multiple of one row to some other row, that's what you are doing in the forward elimination steps, and that's how you end up with a . . . with an upper triangular matrix. Now, the upper triangular matrix, as you can see, will have . . . will have no . . . will have 0 elements below the diagonal, and in that case then the determinant of the upper triangular matrix is simply the multiplication of the diagonal elements.
So you are using these two theorems of determinants of a matrix to be able to find the determinant of a matrix in a simple scientific fashion, that you take the coefficient matrix which is given to you, apply the steps of forward elimination of Naive Gaussian method to get the upper triangular matrix, and since you are only adding the multiples of one row to another row to be able to get to the U matrix, that the determinant of this matrix is going to same as the determinant of this matrix here. But the determinant of an upper triangular matrix is very easy to find, because it's simply the multiple of all the diagonal elements.
So I'm going to write down these theorems, and that turns out to be the background for finding out the determinant of the matrix.
The theorem one says as follows, let's suppose if you have an A, which is an n-by-n matrix, and you end up with a B matrix, and the way you get from an n-by-n matrix to a . . . to a B matrix, which is also n-by-n, is by adding or subtracting a multiple of one row to another row, then the determinant of the two matrices, of A is same as the determinant of the B matrix. So if you are going from this matrix here to another matrix, B, simply by adding or subtracting a multiple of one row to another row, not to itself, but to another row, then your determinant of the matrix does not change. So you're using that as one theorem to be able to find out the determinant of a matrix.
The other theorem which . . . which we are using once we are able to convert our coefficient matrix by using these multiples of one row being added or subtracted from another row, is that the determinant . . . the determinant of an upper triangular matrix, so if you have an upper triangular matrix, let's suppose C, which is n-by-n, then the determinant of C is simply equal to c11 times c22 times c33, all the way up to cnn. So what I'm trying to say is that if you have an upper triangular matrix, the determinant of an upper triangular matrix is simply given by the multiplication of all its diagonal elements, which you can write in the compact form as the product multiple of cii, i is equal to 1 to n.
So these two theorems of a determinant of a matrix are used to find out what the determinant of any matrix is, and that will help you to do the things which you need to do when you are trying to find a determinant, you might be using it to figure out whether a particular matrix is invertible, whether a particular matrix is singular, things like that. And that's the end of this segment.