CHAPTER 04.12: UNARY MATRIX OPERATIONS: Theorems on determinants Part 4 of 4

Let’s talk about some more theorems on determinants. This is another theorem that says let A be a n by n upper triangular, lower triangular, or diagonal matrix. So if you have a square matrix A, which is either upper triangular or lower triangular or diagonal, then the determinant of the matrix A is simply the multiplication of all the diagonal elements. So all you have to do is to multiply all the diagonal elements to each other, and you’’ find the determinant of the A matrix. So what that means is that if you were going to expand this it’d be a11 times a22 times all the way up to ann. So you want to multiply all the diagonal elements and that’s what’s going to return our determinant of the matrix.

So let’s suppose we take an example. If somebody says – hey, go ahead and find the determinant of this matrix A. So you have this A matrix and you want to find the determinant of this matrix. Let’s suppose this is 20, 3, 6, 7, 2, 2, and 0, 0, 0 here. So if you have a matrix like that and somebody says find the determinant of this matrix, now you’ll recognize this is an upper triangular matrix because there are zeros in the elements which are below the diagonal. So since this is an upper triangular matrix, then the determinant of A will be simply the multiple of the diagonal elements 20, 7, and 2. So it’d be 20 times 7 times 2 and that’s 280. So we don’t need to do anything else. If you were going to use the minor method, you’ll find that hey it will turn out to be 280. And in fact you can even show it that hey – if you were to apply the minor method to a general upper triangular, lower triangular, or diagonal matrix, then the determinant of the matrix will turn out to be simple the elements which are on the diagonal; the product of those elements. So that’s the end of this segment.