Difference between revisions of "Invertible matrix"
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(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. | ||
| + | # ''A'' is row equivalent to ''I''. | ||
| + | # ''A''x = b has a unique solution, for all b. | ||
| + | # ''A''x = 0 has only the trivial solution of x = 0. | ||
| + | # ''A'' is row equivalent to an upper triangular matrix having non-zero diagonal entries. | ||
[[Category:linear algebra]] | [[Category:linear algebra]] | ||
[[Category:mathematics]] | [[Category:mathematics]] | ||
Revision as of 00:40, May 14, 2010
An invertible matrix A is one that has a corresponding matrix B such that:
- AB = BA = I (I: identity matrix)
A matrix is invertible if and only if it has a nonzero determinant. Only square matrices (n x n) are invertible.
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-1.
There are five conditions that are equivalent to a matrix A being invertible:
- The determinant of A is not zero.
- A is row equivalent to I.
- Ax = b has a unique solution, for all b.
- Ax = 0 has only the trivial solution of x = 0.
- A is row equivalent to an upper triangular matrix having non-zero diagonal entries.