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| [[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. | Not all matrices have a real inverse. Such matrices are [[LinearAlgebra/MatrixProperties#Interible|non-invertible]]. Calculation of the [[LinearAlgebra/Determinants|determinant]] is the common test for invertibility. if ''|'''A'''| != 0'', then '''''A''''' is invertible. Conversely, if ''|'''A'''| = 0'', then '''''A''''' is non-invertible. Only square matrices can be invertible. However, a non-square matrix can separably have distinct '''left inverse''' and '''right inverse''' matrices. Generally, if ''a < b'', then a matrix with shape ''a'' by ''b'' and rank of ''a'' can have a right inverse; a matrix with shape ''b'' by ''a'' and rank of ''a'' can have a left inverse. By contrast, an invertible square matrix has a true inverse that works on either side: '''''AA'''^-1^ = '''A'''^-1^'''A''' = '''I'''''. ---- |
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| === Properties === | == Properties == |
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| 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. | The core principle of inversions is that a matrix '''''A''''' can be canceled out from a larger system: ''x'''AA'''^-1^ = x''. |
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| An invertible matrix has only one vector in the [[LinearAlgebra/NullSpaces|null space]]: the zero vector. | An invertible matrix has only one vector in the [[LinearAlgebra/NullSpaces|null space]]: the zero vector. If any basis vector of a matrix is a linear transformation of another, then the matrix does not have [[LinearAlgebra/Basis|basis]] and must be non-invertible. |
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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''''' |
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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'''''). {{{ -1 A A = I ┌ ┐┌ ┐ ┌ ┐ │ 1 3││ a b│ │ 1 0│ │ 2 7││ c d│ = │ 0 1│ └ ┘└ ┘ └ ┘ }}} 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 elimination proceeds as: |
Because '''''AA'''^-1^ = '''I''''', applying [[LinearAlgebra/Elimination|elimination]] and [[LinearAlgebra/Elimination#Reduced_Row_Echelon_Form|backwards elimination]] on '''''A''''' augmented with an [[LinearAlgebra/SpecialMatrices#Identity_Matrix|identity matrix]] ('''''I''''') will create '''''A'''^-1^'' in the augmentation. |
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| }}} The reverse elimination proceeds as: {{{ |
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| ---- == Determinant and Cofactor Matrix == Given the [[LinearAlgebra/Determinants|determinant]] of '''''A''''', it can also be simple to compute '''''A'''^-1^'' as ''(1/|'''A'''|)'''C'''^T^''. '''''C''''' is the cofactor matrix, where ''c,,i j,,'' is the cofactor of ''a,,i j,,''. For example, given a 2 x 2 '''''A''''' like: {{{ ┌ ┐ │ a b│ │ c d│ └ ┘ }}} The cofactor matrix '''''C''''' is: {{{ ┌ ┐ │ d -c│ │ -b a│ └ ┘ }}} But this must be transposed to '''''C'''^T^'': {{{ ┌ ┐ │ d -b│ │ -c a│ └ ┘ }}} And then '''''A'''^-1'' is: {{{ ┌ ┐ │ (1/det A) * d (1/det A) * -b│ │ (1/det A) * -c (1/det A) * a│ └ ┘ }}} The above example fits into this formula. The [[LinearAlgebra/Elimination|elimination]] and [[LinearAlgebra/Elimination#Reduced_Row_Echelon_Form|backwards elimination]] prove that the determinant of that '''''A''''' is 1. The more fundamental formula ''ad - bc'' expands to ''1 * 7 - 2 * 3'' which also reveals a determinant of 1. As such, ''(1/|'''A'''|)'' is trivially 1. So simply plug the given (''a'', ''b'', ''c'', ''d'') into the transposed cofactor matrix to find the inverse. |
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 a real inverse. Such matrices are non-invertible.
Calculation of the determinant is the common test for invertibility. if |A| != 0, then A is invertible. Conversely, if |A| = 0, then A is non-invertible.
Only square matrices can be invertible. However, a non-square matrix can separably have distinct left inverse and right inverse matrices. Generally, if a < b, then a matrix with shape a by b and rank of a can have a right inverse; a matrix with shape b by a and rank of a can have a left inverse. By contrast, an invertible square matrix has a true inverse that works on either side: AA-1 = A-1A = I.
Properties
The core principle of inversions is that a matrix A can be canceled out from a larger system: xAA-1 = x.
An invertible matrix has only one vector in the null space: the zero vector. If any basis vector of a matrix is a linear transformation of another, then the matrix does not have basis and must be non-invertible.
For orthogonal matrices (such as permutation matrices), the inverse is also the transpose: Q-1 = QT.
Calculation
Because AA-1 = I, applying elimination and backwards elimination on A augmented with an identity matrix (I) will create A-1 in the augmentation.
┌ ┐ │ [1] 3 │ 1 0│ │ 2 7 │ 0 1│ └ ┘ ┌ ┐ │ [1] 3 │ 1 0│ │ 0 [1] │ -2 1│ └ ┘ ┌ ┐ │ 1 3 │ 1 0│ │ 0 [1] │ -2 1│ └ ┘ ┌ ┐ │ [1] 0 │ 7 -3│ │ 0 [1] │ -2 1│ └ ┘
A-1 is:
┌ ┐ │ 7 -3│ │ -2 1│ └ ┘
Determinant and Cofactor Matrix
Given the determinant of A, it can also be simple to compute A-1 as (1/|A|)CT. C is the cofactor matrix, where ci j is the cofactor of ai j.
For example, given a 2 x 2 A like:
┌ ┐ │ a b│ │ c d│ └ ┘
The cofactor matrix C is:
┌ ┐ │ d -c│ │ -b a│ └ ┘
But this must be transposed to CT:
┌ ┐ │ d -b│ │ -c a│ └ ┘
And then A^-1 is:
┌ ┐ │ (1/det A) * d (1/det A) * -b│ │ (1/det A) * -c (1/det A) * a│ └ ┘
The above example fits into this formula. The elimination and backwards elimination prove that the determinant of that A is 1. The more fundamental formula ad - bc expands to 1 * 7 - 2 * 3 which also reveals a determinant of 1. As such, (1/|A|) is trivially 1. So simply plug the given (a, b, c, d) into the transposed cofactor matrix to find the inverse.