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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^''. | 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^''. |
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.
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│ └ ┘