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| = Inverse Matrices = | = Matrix Inversion = |
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| == Introduction == | 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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| 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 == 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. 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
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.
Contents
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.
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│ └ ┘