Attached matrix

Matrix attached . It is the square matrix of order N that results from transposing the algebraic complements of another square matrix of the same order.

Since the product of the original matrix and its attachment generate a diagonal matrix, whose elements are the determinant of the original matrix, it is used to obtain the inverse matrix .

Summary

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• 1 Definition
• 2 Obtaining attachments of matrices of order 2 and 3
• 3 Properties
• 4 Examples
• 5 Importance
• 6 See also
• 7 Sources

Definition

Let A be a square matrix of order N, it is called the attached matrix of A and the matrix is symbolized by A ⁺ :

where a ⁺ _{i, j} = (- 1) ^{j + i} M _{j, i} are the transposed algebraic complements corresponding to each element a _{j, i} of A such that the smallest M _{i, j} is defined by the determinant :

Obtaining attachments of matrices of order 2 and 3

In the case of square matrices of order 2 and 3 the mechanism is simpler than in those of higher order from 4, although the same idea of ​​the definition is followed.

In the case of A if it has order 2:

His deputy would be:

,

For a matrix A of order 3 it is a little more elaborate:

The algebraic complements of each element of A are proposed as follows:

This would be the transposition of the attached matrix. Then rows are exchanged for columns to get A ⁺ :

Properties

Let be a square matrix A of order N and its adjunct A ⁺ is then true:

1. (kA) ⁺= k ^N A ⁺
2. (AB) ⁺= A ⁺ B ⁺
3. AA ⁺= A ⁺ A = | A | I

Examples

Let be the matrix:

Through the method of obtaining expressed above A ⁺ is:

In the case of the order 3 matrix:

To calculate your A ⁺ , proceed in the same way:

Importance

The property 3 seen before allows from the attached matrix to obtain the inverse matrix by means of the dot product:

A ^-1 = | A | ^-1 A ⁺

Although the Jordan method will be preferred because for order matrices greater than 3, it has fewer iterations and less complexity in calculations and notation.