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<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> | <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> | ||
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| + | Example. | ||
| + | <br> | ||
| + | <br> | ||
| + | <br> | ||
| + | <math>A = \left(\begin{array}{cccc}1&2&3&4\\5&6&7&8\end{array}\right)</math> | ||
<h3> Theorem 1 </h3> | <h3> Theorem 1 </h3> | ||
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<h3> Theorem 2 </h3> | <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>. | + | <p> If <i>A</i> and <i>B</i> are both nonsingular <i>n</i> x <i>n</i> matrices (i.e. invertible), 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 06:50, 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 = In. We call B the inverse of A and denote it as A-1. Thus, AA-1 = A-1A = In. In this case, A is also called nonsingular.
Example.
$ A = \left(\begin{array}{cccc}1&2&3&4\\5&6&7&8\end{array}\right) $
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 (i.e. invertible), then AB is nonsingular and (AB)-1 = B-1A-1.
