Matrix inverse: Difference between revisions
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imported>Richard Pinch (comment to take care of determinant over a ring) |
imported>Richard Pinch ("invertible" is an anchor; inverse is unique) |
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:<math>\mathbf{AX} = \mathbf{XA} = \mathbf{I}_n \ </math> | :<math>\mathbf{AX} = \mathbf{XA} = \mathbf{I}_n \ </math> | ||
('''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'''). | ('''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 its [[determinant]] is itelf 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. | A matrix is '''invertible''' if and only if it possesses an inverse. A matrix is invertible if and only if its [[determinant]] is itelf 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 02:06, 10 December 2008
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. A matrix is invertible if and only if its determinant is itelf invertible: over a field such as the real or complex numbers, this is equivalent to specifying that the determinant does not equal zero.