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| − | <br> | + | ''<br>'' |
== '''Inverse of a Matrix''' == | == '''Inverse of a Matrix''' == | ||
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*Ax=0 has a unique solution | *Ax=0 has a unique solution | ||
*There is a B matrix such that A B = In | *There is a B matrix such that A B = In | ||
| − | *Ax=b has a unique solution for any b---x=A | + | *Ax=b has a unique solution for any b---x=<span class="texhtml">''A''<sup> − 1</sup></span> |
| − | <br> | + | <br> <br> |
== Properties == | == Properties == | ||
| − | *(AB) | + | *<span class="texhtml">(AB)<sup> − 1</sup></span> = <span class="texhtml">''B''<sup> − 1</sup>''A''<sup> − 1</sup></span> |
| + | *(A1 A2.....Ar)<span class="texhtml"><sup> − 1</sup></span>=<span class="texhtml">''A''r'''''<b><sup> − 1</sup></b>'''''A''''''''r'' − 1<sup> − 1</sup>...''A''1<sup> − 1</sup></span> | ||
| + | *<span class="texhtml">(''A''<sup> − 1</sup>)<sup> − 1</sup> = ''A''</span><span class="texhtml"><sup></sup></span> | ||
| + | *<span class="texhtml">(''A''<sup> − 1</sup>)<sup>''T''</sup> = (''A''<sup>''T''</sup>)<sup> − 1</sup></span> | ||
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| − | <br> <br> <br> <br> <math>\left(\begin{array}{cccc}1&2&3&4\\5&6&7&8\end{array}\right)</math> | + | Calculations<br> <br> <br> <br> <math>\left(\begin{array}{cccc}1&2&3&4\\5&6&7&8\end{array}\right)</math> |
Revision as of 14:02, 13 December 2011
Inverse of a Matrix
Definition: Let A be a square matrix of order n x n(square matrix). If there exists a matrix B such that
Then B is called the inverse matrix of A.
Conditions
A n x n is invertible (non-singular) if:
- Ax=0 has a unique solution
- There is a B matrix such that A B = In
- Ax=b has a unique solution for any b---x=A − 1
Properties
- (AB) − 1 = B − 1A − 1
- (A1 A2.....Ar) − 1=Ar − 1A'''r − 1 − 1...A1 − 1
- (A − 1) − 1 = A
- (A − 1)T = (AT) − 1
Calculations
$ \left(\begin{array}{cccc}1&2&3&4\\5&6&7&8\end{array}\right) $
