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<br>
 
<br>
 
<br>
 
<br>
A = <math>\left(\begin{array}{cccc}4&3\\3&2\end{array}\right)</math>  
+
<math>A = \left(\begin{array}{cccc}4&3\\3&2\end{array}\right)</math>  
 
<br>
 
<br>
A<sup>-1</sup> = <math>\left(\begin{array}{cccc}-2&3\\3&-4\end{array}\right)</math>   
+
<math>A<sup>-1</sup> = \left(\begin{array}{cccc}-2&3\\3&-4\end{array}\right)</math>   
  
 
<br>
 
<br>
  
AA<sup>-1</sup> = <math>\left(\begin{array}{cccc}4&3\\3&2\end{array}\right)</math><math>\left(\begin{array}{cccc}-2&3\\3&-4\end{array}\right)</math> = <math>\left(\begin{array}{cccc}1&0\\0&1\end{array}\right)</math>  
+
<math>AA<sup>-1</sup> = \left(\begin{array}{cccc}4&3\\3&2\end{array}\right)</math><math>\left(\begin{array}{cccc}-2&3\\3&-4\end{array}\right) = </math> <math>\left(\begin{array}{cccc}1&0\\0&1\end{array}\right)</math>  
  
 
<br>
 
<br>
  
and A<sup>-1</sup>A = </math><math>\left(\begin{array}{cccc}-2&3\\3&-4\end{array}\right)</math><math>\left(\begin{array}{cccc}4&3\\3&2\end{array}\right)</math> = <math>\left(\begin{array}{cccc}1&0\\0&1\end{array}\right)</math>   
+
<math>and A<sup>-1</sup>A = \left(\begin{array}{cccc}-2&3\\3&-4\end{array}\right)</math><math>\left(\begin{array}{cccc}4&3\\3&2\end{array}\right) = </math><math>\left(\begin{array}{cccc}1&0\\0&1\end{array}\right)</math>   
  
  

Revision as of 07:59, 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}4&3\\3&2\end{array}\right) $
$ A<sup>-1</sup> = \left(\begin{array}{cccc}-2&3\\3&-4\end{array}\right) $


$ AA<sup>-1</sup> = \left(\begin{array}{cccc}4&3\\3&2\end{array}\right) $$ \left(\begin{array}{cccc}-2&3\\3&-4\end{array}\right) = $ $ \left(\begin{array}{cccc}1&0\\0&1\end{array}\right) $


$ and A<sup>-1</sup>A = \left(\begin{array}{cccc}-2&3\\3&-4\end{array}\right) $$ \left(\begin{array}{cccc}4&3\\3&2\end{array}\right) = $$ \left(\begin{array}{cccc}1&0\\0&1\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.

Alumni Liaison

Correspondence Chess Grandmaster and Purdue Alumni

Prof. Dan Fleetwood