Inverse matrix

Inverse matrix . Said of squareMatrix A ^-1 of order N that given a nonsingular A square matrix of the same order, satisfies AA ^-1 = A ^-1 A = I _N .

Summary

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• 1
• 2
• 3 Methods of obtaining inverse matrices
• 4 Solution of systems of linear equations using the inverse
• 5
  • 1 Solving a system of linear equations
• 6 See also
• 7 Sources

Definitions.

Given the square matrix A of order N it is said to be an invertible matrix if and only if the determinant of A is not null.

If A is an invertible matrix, it is said that the square matrix A ^-1 of order N is the inverse matrix A such that AA ^-1 = A ^-1 A = I _N .

Properties.

Relating to invertibility:

• A square matrix Ais invertible if there is another square matrix B such that AB = BA = I.
• An array is invertible ssi:
  • It’s squared.
  • It is non-singular.
• If a matrix Aof order n is invertible, then its range is also n .
• The inverse matrix if it exists is unique.
• (AB) ^-1= A ^-1 B ^-1
• (A ^T) ^-1 = (A ^-1 ) ^T
• (A ^-1) ^-1 = A
• I ^-1= I

Methods of obtaining inverse matrices

Using the determinant and the attached matrix:

• Let A ^{+ be}the attached matrix of A , A ^-1 = A ⁺ / | A | .

For the specific cases of invertible matrices of order 2 and 3, their inverses can be calculated in the following ways:

• Order 2:
  • ,
  • Whose determinate is | A | = ad-bc,
  • ,
  • ,
• Order 3:
  • ,
  • Its determinant
  • The algebraic complements of each cell are calculated, obtaining the transposition of the attached matrix ,
  • It transposes: ,

Gauss-Jordan Elimination Method :

• From, the extended matrix (A | I) is createdand through elementary transformations in each row, if A is invertible, it is obtained (I | A ^-1 ) .

Solution of systems of linear equations using the inverse

Let the matrix A be non-singular and the system of linear equations represented in the matrix equation:

• AX = B

It can be multiplied in front of both sides of the equation by A ^-1 :

• A ^-1AX = A ^-1 B => IX = A ^-1 B

from where:

• X = A ^-1B

which shows another way to determine the solution of a compatible linear equation system.

Examples.

Matrix:

it has no inverse because its determinant is 0, that is, it is a singular matrix.

Instead matrix A :

complies with | A | = -2 , is non-singular and therefore is invertible. A ^-1 is:

Which can be obtained through the attached matrix and its determinant:

Or by the Jordan method:

• Write the augmented matrix (A | I): .
• Pivot A ₁.
  • .
  • Replace Row2 with Row2 – 3 Row1.
  • Replace Row3 with Row3 – 5 Row1.
• Pivot A ₂.
  • .
  • Replace Row1 with Row1 + Row2.
  • Replace Row3 with Row3 – 2 Row2.
• Pivot A ₃.
  • .
  • Replace Row1 with Row1 + Row3.
  • Substitute Row2 for Row2 + 4 Row3.
• Row 2 is divided by -2 and finally (I | A ^-1) is obtained :
  • .

Linear equation system resolution

Let be the system of linear equations represented by the matrix equation AX = B :

But the matrix representing the SEL coefficients has the inverse:

And in this case the system can be restated as X = A ^-1 B , solving by means of the matrix product: