Complementary minor and attachment of an element. Attached Matrix.

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:

minor complementary

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:

minor complementary to an element

And the smallest complementary will be the determinant that results from eliminating that row and column.

minor complementary to an array

Let’s get the smallest complementary of 2.

We remove the row and the column where 2 is:

attachment of an element

The remaining elements form the smallest complementary of 2:

attached minor

We do the same with 3:

minor complementary and attached

The smallest complementary of 3 is:

minor of a matrix

And finally, we remove the row and the column where the 5 is to find its least complementary:

element of an array

Attachment of a matrix

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.

elements of an array examples

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:

minor complementary matrices

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).

Attachments of a matrix

We now operate:

minor of a matrix

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:

complementary minors

adjunto matriz

The 3 is in row 1 and in column 3, therefore, the -1 will be raised to 1+3 and its attachment will be:

main minor of a matrix

adjunto matematicas

The 5 is in row 2 and column 2. The attachment of 5 will be:

elements of a matrix definition

which is the attachment of a matrix

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.

Attached matrix

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:

complementary element

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:

determinant of the attached matrix

To complete the attached matrix of A, we will continue to calculate the attachments of 4, 6, 7, 8 and 9.

Attachment of 4:

complementary matrix

Attachment of 6:

attachments matrix

Attachment of 7:

matrix adjunta

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.

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