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<h3> Theorem 3 </h3>
 
<h3> Theorem 3 </h3>
  
<p> If <i>A</i> is a nonsingular matrix, then <i>A</i><sup>-1</sup> is nonsingular and (<i>A</i><sup>-1</sup>)<sup>-1</sup> = <i>A</i>.
+
<p> If <i>A</i> is a nonsingular matrix, then <i>A</i><sup>-1</sup> is nonsingular and (<i>A</i><sup>-1</sup>)<sup>-1</sup> = <i>A</i>. </p>
  
 +
<h3> Theorem 4 </h3>
  
 +
<p> If A is a nonsingular matrix, then <i>A</i><sup>T</sup> is nonsingular and (<i>A</i><sup>-1</sup>)<sup>T</sup> = (<i>A</i><sup>T</sup>)<sup>-1</sup>.
  
  

Revision as of 07:08, 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.

Theorem 3

If A is a nonsingular matrix, then A-1 is nonsingular and (A-1)-1 = A.

Theorem 4

If A is a nonsingular matrix, then AT is nonsingular and (A-1)T = (AT)-1.

Alumni Liaison

Prof. Math. Ohio State and Associate Dean
Outstanding Alumnus Purdue Math 2008

Jeff McNeal