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<h1> Inverse of a Matrix </h1> | <h1> Inverse of a Matrix </h1> | ||

− | <p> An <i>n</i> x <i>n</i> matrix <i>A</i> is said to have an inverse provided there exists an <i>n</i> x <i>n</i> matrix <i>B</i> such that <i>AB</i> = <i>BA</i> = <i>I</i><sub>n</sub>. We call <i>B</i> the inverse of <i>A</i> and denote it as <i>A</i><sup>-1</sup>. Thus, <i>A</i><i>A</i><sup>-1</sup> = <i>A</i><sup>-1</sup><i>A</i> = <i>I</i><sub>n</sub>. In this case, A is also called nonsingular. | + | <p> An <i>n</i> x <i>n</i> matrix <i>A</i> is said to have an inverse provided there exists an <i>n</i> x <i>n</i> matrix <i>B</i> such that <i>AB</i> = <i>BA</i> = <i>I</i><sub>n</sub>. We call <i>B</i> the inverse of <i>A</i> and denote it as <i>A</i><sup>-1</sup>. Thus, <i>A</i><i>A</i><sup>-1</sup> = <i>A</i><sup>-1</sup><i>A</i> = <i>I</i><sub>n</sub>. In this case, A is also called nonsingular. </p> |

+ | <h3> Theorem 1 </h3> | ||

+ | <p> The inverse of a matrix, if it exists, is unique </p> | ||

+ | <h3> Theorem 2 </h3> | ||

+ | |||

+ | <p> If <i>A</i> and <i>B</i> are both nonsingular <i>n</i> x <i>n</i> matrices, then <i>AB</i> is nonsingular and (<i>AB</i><sup>)-1</sup> = <i>B</i><sup>-1</sup><i>A</i><sup>-1</sup>. | ||

## Revision as of 06:46, 10 December 2012

# Inverse of a Matrix

An *n* x *n* matrix *A* is said to have an inverse provided there exists an *n* x *n* matrix *B* such that *AB* = *BA* = *I*_{n}. We call *B* the inverse of *A* and denote it as *A*^{-1}. Thus, *A**A*^{-1} = *A*^{-1}*A* = *I*_{n}. In this case, A is also called nonsingular.

### Theorem 1

The inverse of a matrix, if it exists, is unique

### Theorem 2

If *A* and *B* are both nonsingular *n* x *n* matrices, then *AB* is nonsingular and (*AB*^{)-1} = *B*^{-1}*A*^{-1}.