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Invertible matrix (view source)

Revision as of 00:40, May 14, 2010

350 bytes added ,  00:40, May 14, 2010
There are five conditions that are equivalent to a matrix ''A'' being invertible:
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It can be shown that, for any invertible matrix ''A'', there is only one matrix ''B'' such that ''AB'' = I; this unique matrix is called the [[inverse matrix]] of ''A'', and represented as ''A<sup>-1</sup>''.
 
It can be shown that, for any invertible matrix ''A'', there is only one matrix ''B'' such that ''AB'' = I; this unique matrix is called the [[inverse matrix]] of ''A'', and represented as ''A<sup>-1</sup>''.
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There are five conditions that are equivalent to a matrix ''A'' being invertible:
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# The determinant of ''A'' is not zero.
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# ''A'' is row equivalent to ''I''.
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# ''A''x = b has a unique solution, for all b.
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# ''A''x = 0 has only the trivial solution of x = 0.
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# ''A'' is row equivalent to an upper triangular matrix having non-zero diagonal entries.
    
[[Category:linear algebra]]
 
[[Category:linear algebra]]
 
[[Category:mathematics]]
 
[[Category:mathematics]]
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