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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.   
 
Example.   
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<h2> Methods for determining the inverse of a matrix </h2>
 
<h2> Methods for determining the inverse of a matrix </h2>
  
<h4> Shortcut for determining the inverse of a 2 x 2 matrix </h4>
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<h4> 1. Shortcut for determining the inverse of a 2 x 2 matrix </h4>
  
 
<p> If      <math>A = \left(\begin{array}{cccc}a&b\\c&d\end{array}\right)</math>  then the inverse of matrix A can be found using:
 
<p> If      <math>A = \left(\begin{array}{cccc}a&b\\c&d\end{array}\right)</math>  then the inverse of matrix A can be found using:
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<br>
 
<br>
 
<br>
 
<br>
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<p> Example  
 
<p> Example  
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<math>A^{-1} = \left(\begin{array}{cccc}-2&1\\\frac{3}{2}&\frac{-1}{2}\end{array}\right)</math>
 
<math>A^{-1} = \left(\begin{array}{cccc}-2&1\\\frac{3}{2}&\frac{-1}{2}\end{array}\right)</math>
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</p>
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<h4> 2. Augmented Matrix Method (Can be used for any n x n matrix) </h4>
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<p> Use Gauss-Jordan Elimination to transform [ A | I ] into [ I | A<sup>-1</sup> ]. </p>
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<math>\left(\begin{array}{cccc}a&b&c\\d&e&f\\g&h&i\end{array}\right)</math>
  
  
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<p> Example
  
  

Revision as of 07:25, 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^{-1} = \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.



Methods for determining the inverse of a matrix

1. Shortcut for determining the inverse of a 2 x 2 matrix

If $ A = \left(\begin{array}{cccc}a&b\\c&d\end{array}\right) $ then the inverse of matrix A can be found using:
$ A^{-1} = \frac{1}{detA}\left(\begin{array}{cccc}d&-b\\-c&a\end{array}\right) = \frac{1}{ad - bc}\left(\begin{array}{cccc}d&-b\\-c&a\end{array}\right) $




Example

$ A = \left(\begin{array}{cccc}1&2\\3&4\end{array}\right) $
$ detA = ad - bc = 1 \times 4 - 2 \times 3 = -2 $
$ A^{-1} = \frac{1}{-2}\left(\begin{array}{cccc}4&-2\\-3&1\end{array}\right) $
$ A^{-1} = \left(\begin{array}{cccc}-2&1\\\frac{3}{2}&\frac{-1}{2}\end{array}\right) $



2. Augmented Matrix Method (Can be used for any n x n matrix)

Use Gauss-Jordan Elimination to transform [ A | I ] into [ I | A-1 ].

$ \left(\begin{array}{cccc}a&b&c\\d&e&f\\g&h&i\end{array}\right) $


Example

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