In this lesson I am going to explain a series of concepts that will help us to calculate the determinant of a matrix, as well as to calculate the inverse of a matrix.
I will show you what they are and how to calculate the smallest complementary of the elements of a matrix, their attachments and the attached matrix.
Complementary minor of an element
For a square matrix, the smallest complementary of an element is the determinant that results from suppressing the row and column where that element is found in the matrix.
For example, we have the following matrix:
We are going to obtain the complementary minors of elements 1, 2, 3 and 5.
In order to obtain the complementary minor of 1, we eliminate the row and the column where 1 is located:
And the smallest complementary will be the determinant that results from eliminating that row and column.
Let’s get the smallest complementary of 2.
We remove the row and the column where 2 is:
The remaining elements form the smallest complementary of 2:
We do the same with 3:
The smallest complementary of 3 is:
And finally, we remove the row and the column where the 5 is to find its least complementary:
More than with the name of the concept, stay how it works.
Once we have the smallest complementary, we can calculate the attachment of an element, which is the concept I’ll explain below:
Adjustment of an element
For a square matrix, the attachment of an element in the matrix is equal to the smallest complementary element, multiplied by -1 elevated to the sum of the row plus the column where that element is found in the matrix.
i and j are the numbers of the row and column respectively where the element is located.
Let’s see it with several examples to make it much clearer. For example, for the same matrix of the previous example:
Let’s calculate the attachment of 1.
The 1 is in row 1 and in column 1, therefore, to calculate the attachment of 1, the smallest complementary of 1, we have to multiply it by -1 elevated 1+1 (row 1 and column 1).
We now operate:
And we already have the attachment of 1
To calculate the attachment of 2, we do the same thing, only now, -1 will be raised to 3 (1+2), since 2 is in row 1 and column 2:
The 3 is in row 1 and in column 3, therefore, the -1 will be raised to 1+3 and its attachment will be:
The 5 is in row 2 and column 2. The attachment of 5 will be:
The attachment of an element allows us to calculate both the value of a determinant and to calculate the attached matrix, which is what we will see next.
The attached matrix of another matrix, is a matrix where each of the elements are replaced by their attachments.
We are going to calculate the attached matrix of matrix A:
In order to obtain the attached matrix, the elements are replaced by each of their attachments.
In the previous section we have calculated the attachments of 1, 2, 3 and 5:
- Adjunt of 1 = -3
- Attachment of 2 = 6
- Adjunt of 3 = -3
- Attachment of 5 = -12
These attachments can now be replaced by the elements:
To complete the attached matrix of A, we will continue to calculate the attachments of 4, 6, 7, 8 and 9.
Attachment of 4:
Attachment of 6:
Attachment of 7:
Attached of 8:
Attachment of 9:
Once we have all the attachments, we can replace them with their elements and the attached matrix of A will be:
The attached matrix will help us to calculate the inverse of an array.