Difference between revisions of "Inverse matrix"
From Encyclopedia of Mathematics
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''of a square matrix $A$ over a field $k$'' | ''of a square matrix $A$ over a field $k$'' | ||
| − | The matrix $A^{-1}$ for which $AA^{-1}=A^{-1}A=E$, where $E$ is the identity matrix. Invertibility of a matrix is equivalent to its being non-singular (see [[ | + | The matrix $A^{-1}$ for which $AA^{-1}=A^{-1}A=E$, where $E$ is the identity matrix. Invertibility of a matrix is equivalent to its being non-singular (see [[Non-singular matrix]]). For the matrix $A=\|\alpha_{ij}\|$, the inverse matrix is $A^{-1}=\|\gamma_{ij}\|$ where |
$$\gamma_{ij}=\frac{A_{ji}}{\det A},$$ | $$\gamma_{ij}=\frac{A_{ji}}{\det A},$$ | ||
| − | where $A_{ij}$ is the [[ | + | where $A_{ij}$ is the [[cofactor]] of the element $\alpha_{ij}$. For methods of computing the inverse of a matrix see [[Inversion of a matrix]]. |
Latest revision as of 18:09, 14 December 2015
2020 Mathematics Subject Classification: Primary: 15A09 [MSN][ZBL]
of a square matrix $A$ over a field $k$
The matrix $A^{-1}$ for which $AA^{-1}=A^{-1}A=E$, where $E$ is the identity matrix. Invertibility of a matrix is equivalent to its being non-singular (see Non-singular matrix). For the matrix $A=\|\alpha_{ij}\|$, the inverse matrix is $A^{-1}=\|\gamma_{ij}\|$ where
$$\gamma_{ij}=\frac{A_{ji}}{\det A},$$
where $A_{ij}$ is the cofactor of the element $\alpha_{ij}$. For methods of computing the inverse of a matrix see Inversion of a matrix.
How to Cite This Entry:
Inverse matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Inverse_matrix&oldid=36931
Inverse matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Inverse_matrix&oldid=36931