Line 27: Line 27:
 
<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 (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>.  
+
<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>.</p>
 +
 
 +
<h3> Corollary 1 </h3>
 +
 
 +
<p> If <i>A</i><sub>1</sub>, <i>A</i><sub>2</sub>, ..., <i>A</i><sub>r</sub> are <i>n</i> x <i>n</i> nonsingular matrices, then <i>A</i><sub>1</sub><i>A</i><sub>2</sub>...<i>A</i><sub>r</sub> is nonsingular an (<i>A</i><sub>1</sub><i>A</i><sub>2</sub>...<i>A</i><sub>r</sub>)<sup>-1</sup> = <i>A</i><sub>r</sub><sup>-1</sup><i>A</i><sub>r-1</sub><sup>-1</sup>...<i>A</i><sub>1</sub><sup>-1</sup>. </p>
 +
 
  
  

Revision as of 07:05, 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-1 = $ \left(\begin{array}{cccc}-2&3\\3&-4\end{array}\right) $


AA-1 = $ \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-1A = $ \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.

Corollary 1

If A1, A2, ..., Ar are n x n nonsingular matrices, then A1A2...Ar is nonsingular an (A1A2...Ar)-1 = Ar-1Ar-1-1...A1-1.

Alumni Liaison

Basic linear algebra uncovers and clarifies very important geometry and algebra.

Dr. Paul Garrett