Matrix inverse: Difference between revisions
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In [[matrix algebra]], the '''inverse''' of a [[square matrix]] '''A''' is '''X''' if | |||
:<math>\mathbf{AX} = \mathbf{XA} = \mathbf{I}_n \ </math> | :<math>\mathbf{AX} = \mathbf{XA} = \mathbf{I}_n \ </math> | ||
where '''I'''<sub>''n''</sub> is the ''n''-by-''n'' [[identity matrix]] | |||
A matrix is '''invertible''' if and only if it possesses an inverse. | If this equation is true, '''X''' is the inverse of '''A''', denoted by '''A'''<sup>-1</sup>. '''A''' is also the inverse of '''X'''. | ||
A matrix is '''invertible''' if and only if it possesses an inverse. | |||
=== Uniqueness === | |||
Every invertible matrix has only one inverse. | |||
For example, if '''AX''' = '''I''' and '''AY''' = '''I''', then '''X''' = '''Y'''. | |||
So, '''X''' = '''Y''' = '''A'''<sup>-1</sup>. | |||
To prove this, consider the case of '''X'''.'''A'''.'''Y'''. | |||
=== Calculation === | |||
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.[[Category:Suggestion Bot Tag]] | |||
Latest revision as of 16:01, 16 September 2024
In matrix algebra, the inverse of a square matrix A is X if
where In is the n-by-n identity matrix
If this equation is true, X is the inverse of A, denoted by A-1. A is also the inverse of X.
A matrix is invertible if and only if it possesses an inverse.
Uniqueness
Every invertible matrix has only one inverse.
For example, if AX = I and AY = I, then X = Y. So, X = Y = A-1.
To prove this, consider the case of X.A.Y.
Calculation
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.