Matrix inverse: Difference between revisions
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imported>Richard Pinch (refine link, give context (unfortunately red)) |
imported>Richard Pinch (refer to adjugate) |
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('''I'''<sub>''n''</sub> is the ''n''-by-''n'' [[identity matrix]]). If this equation is true, '''X''' is the inverse of '''A''', denoted by '''A'''<sup>-1</sup> ( and '''A''' is the inverse of '''X'''). Furthermore, the inverse '''X''' is unique (if '''Y''' were also an inverse consider '''X'''.'''A'''.'''Y'''). | ('''I'''<sub>''n''</sub> is the ''n''-by-''n'' [[identity matrix]]). If this equation is true, '''X''' is the inverse of '''A''', denoted by '''A'''<sup>-1</sup> ( and '''A''' is the inverse of '''X'''). Furthermore, the inverse '''X''' is unique (if '''Y''' were also an inverse consider '''X'''.'''A'''.'''Y'''). | ||
A matrix is '''invertible''' if and only if it possesses an inverse. | A matrix is '''invertible''' if and only if it possesses an inverse. The inverse may be computed from the [[adjugate matrix]], which shows that a matrix is invertible if and only if its [[determinant]] is itself invertible: over a [[field (mathematics)|field]] such as the [[real number|real]] or [[complex number]]s, this is equivalent to specifying that the determinant does not equal zero. | ||
Revision as of 15:21, 10 December 2008
In matrix algebra, the inverse of a square matrix A is X if
(In is the n-by-n identity matrix). If this equation is true, X is the inverse of A, denoted by A-1 ( and A is the inverse of X). Furthermore, the inverse X is unique (if Y were also an inverse consider X.A.Y).
A matrix is invertible if and only if it possesses an inverse. The inverse may be computed from the adjugate matrix, which shows that a matrix is invertible if and only if its determinant is itself invertible: over a field such as the real or complex numbers, this is equivalent to specifying that the determinant does not equal zero.