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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]]. The calculation of an inverse matrix, if it exists, is called '''inversion'''. |
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| == Introduction == | == Definition == |
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| An '''inverse matrix''' is a matrix A^-1^ where multiplying it by matrix A results in the identity matrix. | An inverse matrix satisfies the equation '''''AA'''^-1^ = '''I'''''. |
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| If A^-1^ exists, then A is '''invertible''' and '''non-singular'''. Not all matrices are invertible. ---- |
[[LinearAlgebra/Determinants|Determinants]] are the test for '''invertability'''. if ''|'''A'''| != 0'', then '''''A''''' is invertable and non-singular. Conversely, if ''|'''A'''| = 0'', then '''''A''''' is singular and non-invertable. |
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| == Notable Properties == | === Properties === |
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| For a permutation matrix, the inverse is also the 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 [[LinearAlgebra/Orthogonality#Matrices|orthogonal matrices]] (such as [[LinearAlgebra/SpecialMatrices#Permutation_Matrices|permutation matrices]]), the inverse is also the [[LinearAlgebra/MatrixTransposition|transpose]]: '''''Q'''^-1^ = '''Q'''^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 problem: | 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.
Determinants are the test for invertability. if |A| != 0, then A is invertable and non-singular. Conversely, if |A| = 0, then A is singular and non-invertable.
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 orthogonal matrices (such as permutation matrices), the inverse is also the transpose: Q-1 = QT.
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│ └ ┘