Difference between revisions of "Matrix inversion"
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Prerequesite for the inversion is, that <math>\mathbf{A}</math> is an n-by-n square matrix and that <math>\mathbf{A}</math> is regular. Regular means that the row and column vectors are linearly independent and so the determinant is nonzero:<br/><br/> | Prerequesite for the inversion is, that <math>\mathbf{A}</math> is an n-by-n square matrix and that <math>\mathbf{A}</math> is regular. Regular means that the row and column vectors are linearly independent and so the determinant is nonzero:<br/><br/> | ||
<math>det(\mathbf{A})\ne0</math><br/><br/> | <math>det(\mathbf{A})\ne0</math><br/><br/> | ||
− | Otherwise the matrix is called singular. | + | Otherwise the matrix is called singular. <br/> |
− | Before determining the inverse of a matrix it is always useful to compute the determinant and check whether the matrix is regular or singular. If it is singular it is not possible to determine the inverse because there is no inverse. For 3-by-3 and smaller matrices there are simple formulas to compute the determinant. To compute the determinant of larger matrices the | + | Before determining the inverse of a matrix it is always useful to compute the determinant and check whether the matrix is regular or singular. If it is singular it is not possible to determine the inverse because there is no inverse. For 3-by-3 and smaller matrices there are simple formulas to compute the determinant. To compute the determinant of larger matrices the first article explains the minors and cofactors of a matrix on which the an example formula for a 4-by-4 matrix is based, that is presented in the second subarticle. The last two subarticles describe two of the common procedures to determine the inverse of a matrix. |
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# [[Minors and cofactors]] | # [[Minors and cofactors]] |
Revision as of 15:30, 9 May 2014
This article describes the inversion of matrices. It starts with a short explanation of what the inverse of a matrix actually is. Afterwards subarticles present some matrix features and two different approaches to compute the inverse of a matrix based on these features.
The inverse of an n-by-n square matrix is denoted as and defined such that
where is the n-by-n identity matrix.
Prerequesite for the inversion is, that is an n-by-n square matrix and that is regular. Regular means that the row and column vectors are linearly independent and so the determinant is nonzero:
Otherwise the matrix is called singular.
Before determining the inverse of a matrix it is always useful to compute the determinant and check whether the matrix is regular or singular. If it is singular it is not possible to determine the inverse because there is no inverse. For 3-by-3 and smaller matrices there are simple formulas to compute the determinant. To compute the determinant of larger matrices the first article explains the minors and cofactors of a matrix on which the an example formula for a 4-by-4 matrix is based, that is presented in the second subarticle. The last two subarticles describe two of the common procedures to determine the inverse of a matrix.
- Minors and cofactors
- Computing the determinant of a 4-by-4 matrix
- Gauß-Jordan-Algorithm
- Adjugate Formula
Example: inverse of matrix
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