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. |