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| = Inverse Matrices = | = Matrix Inversion = |
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| == Introduction == | 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. |
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| An '''inverse matrix''' is a matrix A^-1^ where multiplying it by matrix A results in the identity matrix. | <<TableOfContents>> |
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| Consider the below problem: | ---- == Definition == 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]]. Not all matrices have an inverse matrix. 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. 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''''' ---- == Calculation == 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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| -1 A A = 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'''''. | |
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== Gauss-Jordan Calculation == The inverse matrix can be calculated through elimination and reverse elimination. First step: |
The elimination proceeds as: |
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2 - 1m = 0 m = 2 2 7 0 1 - 1m - 3m - 1m - 0m ____ ____ ____ ____ 0 1 -2 1 ┌ ┐ │ [1] 3 │ 1 0│ │ 0 1 │ -2 1│ └ ┘ |
┌ ┐ │ [1] 3 │ 1 0│ │ 0 [1] │ -2 1│ └ ┘ |
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| Second step: | The reverse elimination proceeds as: |
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3 - 1m = 0 m = 3 1 3 1 0 - 0m - 1m - -2m - 1m ____ ____ _____ ____ 1 0 7 -3 ┌ ┐ │ 1 0 │ 7 -3│ │ 0 [1] │ -2 1│ └ ┘ |
┌ ┐ │ [1] 0 │ 7 -3│ │ 0 [1] │ -2 1│ └ ┘ |
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| The inverse matrix of A is: | '''''A'''^-1^'' is: |
Matrix Inversion
Given a matrix A and an inverse matrix A-1, the product is the identity matrix. This is an important property with several utilities.
Contents
Definition
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.
Not all matrices have an inverse matrix. 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.
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│ └ ┘