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= Invertibility = |
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| '''Invertability''' is a property of square matrices. If a matrix is invertible, there is an inverse matrix that it can be multiplied by to produce the [[LinearAlgebra/SpecialMatrices#Identity_Matrix|identity matrix]]. The calculation of an inverse matrix is '''inversion'''. | '''Invertibility''' is a property of square matrices. If a matrix is invertible, there is an inverse matrix that it can be multiplied by to produce the [[LinearAlgebra/SpecialMatrices#Identity_Matrix|identity matrix]]. The calculation of an inverse matrix is '''inversion'''. |
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| If a matrix cannot be inverted, it is '''singular''' and '''degenerate''' and '''non-invertible'''. |
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| == Determinant == | === Properties === |
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| The [[LinearAlgebra/Determinant|determinant]] is the most common test for invertibility. If ''|'''A'''| != 0'', then '''''A''''' is invertible. If ''|'''A'''| = 0'', then '''''A''''' is non-invertible. ---- == Properties == The core principle of inversions is that a matrix '''''A''''' can be canceled out from a larger system: ''x'''AA'''^-1^ = x''. |
By definition, '''''AA'''^-1^ = '''A'''^-1^'''A''' = '''I'''''. |
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| == Calculation == | == Test with Determinant == The [[LinearAlgebra/Determinant|determinant]] is the most common test for invertibility. If ''|'''A'''| != 0'', then '''''A''''' is invertible. If ''|'''A'''| = 0'', then '''''A''''' is non-invertible. ---- == Calculation with Elimination == |
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| == Determinant and Cofactor Matrix == | == Calculation with Determinants and Cofactor Matrices == |
Invertibility
Invertibility is a property of square matrices. If a matrix is invertible, there is an inverse matrix that it can be multiplied by to produce the identity matrix. The calculation of an inverse matrix is inversion.
Contents
Definition
A matrix A is invertible if there is a matrix A-1 which satisfies AA-1 = A-1A = I.
If a matrix cannot be inverted, it is singular and degenerate and 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 m < n, then a matrix with shape m by n and rank of m can have a right inverse; a matrix with shape n by m and rank of m can have a left inverse.
Properties
By definition, AA-1 = A-1A = I.
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.
Test with Determinant
The determinant is the most common test for invertibility. If |A| != 0, then A is invertible. If |A| = 0, then A is non-invertible.
Calculation with Elimination
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│ └ ┘
Calculation with Determinants and Cofactor Matrices
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.