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Given a matrix ''A'' and an inverse matrix ''A^-1^'', the product is the [[LinearAlgebra/SpecialMatrices#Identity_Matrix|identity matrix]]. This is an important property with several utilities. For some matrices '''''A''''', the '''inverse matrix''' ('''''A'''^-1^'') is a matrix which can be multiplied by the original matrix to produce the [[LinearAlgebra/SpecialMatrices#Identity_Matrix|identity matrix]]. The calculation of an inverse matrix, if it exists, is called '''inversion'''.
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For some matrices ''A'', the '''inverse matrix''' (''A^-1^'') is a matrix which can be multiplied by the original matrix to produce the [[LinearAlgebra/SpecialMatrices#Identity_Matrix|identity matrix]]. An inverse matrix satisfies the equation '''''AA'''^-1^ = '''I'''''.
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Not  all matrices have an inverse matrix. If ''A^-1^'' exists, then A is '''invertible''' and '''non-singular'''. Not all matrices have an inverse that can satisfy that condition. If '''''A'''^-1^'' exists, then '''''A''''' is '''invertible''' and '''non-singular'''.
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For a [[LinearAlgebra/SpecialMatrices#Permutation_Matrices|permutation matrix]] ''P'', the inverse is also the [[LinearAlgebra/MatrixTransposition|transpose]]: ''P^-1^ = P^T^''. The core principle of inversions is that a matrix '''''A''''' can be canceled out from a larger equation. '''''AA'''^-1^ = '''I''''', so the two terms cancel out.
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For a square matrix ''A'', the '''left inverse''' is the same as the '''right inverse'''. ''AA^-1^ = A^-1^A = I'' An invertible matrix has only one vector in the [[LinearAlgebra/NullSpaces|null space]]: the zero vector.

For a [[LinearAlgebra/SpecialMatrices#Permutation_Matrices|permutation matrix]] '''''P''''', the inverse is also the [[LinearAlgebra/MatrixTransposition|transpose]]: '''''P'''^-1^ = '''P'''^T^''.

For a square matrix '''''A''''', the '''left inverse''' is the same as the '''right inverse'''. '''''AA'''^-1^ = '''A'''^-1^'''A''' = '''I'''''
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Consider the below system, which shows an unknown matrix (''A^-1^'') multiplied by a known matrix (''A'') creating an [[LinearAlgebra/SpecialMatrices#Identity_Matrix|identity matrix]] (''I''). Consider the below system, which shows an unknown matrix ('''''A'''^-1^'') multiplied by a known matrix ('''''A''''') creating an [[LinearAlgebra/SpecialMatrices#Identity_Matrix|identity matrix]] ('''''I''''').
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The inverse matrix is calculated with [[LinearAlgebra/Elimination|elimination]] and [[LinearAlgebra/Elimination#Reduced_Row_Echelon_Form|reverse elimination]]. [[LinearAlgebra/Elimination#Simplification_with_Augmented_Matrices|Augment]] ''A'' with ''I''. The inverse matrix is calculated with [[LinearAlgebra/Elimination|elimination]] and [[LinearAlgebra/Elimination#Reduced_Row_Echelon_Form|reverse elimination]]. [[LinearAlgebra/Elimination#Simplification_with_Augmented_Matrices|Augment]] '''''A''''' with '''''I'''''.
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''A^-1^'' is: '''''A'''^-1^'' is:

Matrix Inversion

For some matrices A, the inverse matrix (A-1) is a matrix which can be multiplied by the original matrix to produce the identity matrix. The calculation of an inverse matrix, if it exists, is called inversion.


Definition

An inverse matrix satisfies the equation AA-1 = I.

Not all matrices have an inverse that can satisfy that condition. If A-1 exists, then A is invertible and non-singular.

Properties

The core principle of inversions is that a matrix A can be canceled out from a larger equation. AA-1 = I, so the two terms cancel out.

An invertible matrix has only one vector in the null space: the zero vector.

For a permutation matrix P, the inverse is also the transpose: P-1 = PT.

For a square matrix A, the left inverse is the same as the right inverse. AA-1 = A-1A = I


Calculation

Consider the below system, which shows an unknown matrix (A-1) multiplied by a known matrix (A) creating an identity matrix (I).

         -1
  A     A    =   I

┌    ┐┌    ┐   ┌    ┐
│ 1 3││ a b│   │ 1 0│
│ 2 7││ c d│ = │ 0 1│
└    ┘└    ┘   └    ┘

The inverse matrix is calculated with elimination and reverse elimination. Augment A with I.

The elimination proceeds as:

┌            ┐
│ [1] 3 │ 1 0│
│  2  7 │ 0 1│
└            ┘
┌               ┐
│ [1]  3  │  1 0│
│  0  [1] │ -2 1│
└               ┘

The reverse elimination proceeds as:

┌             ┐
│ 1  3  │  1 0│
│ 0 [1] │ -2 1│
└             ┘
┌                ┐
│ [1]  0  │  7 -3│
│  0  [1] │ -2  1│
└                ┘

A-1 is:

┌      ┐
│  7 -3│
│ -2  1│
└      ┘


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LinearAlgebra/MatrixInversion (last edited 2024-06-06 02:58:56 by DominicRicottone)