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## Revision as of 07:28, 10 December 2012

## Contents

# 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* = *I*_{n}. We call *B* the inverse of *A* and denote it as *A*^{-1}. Thus, *A**A*^{-1} = *A*^{-1}*A* = *I*_{n}. 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*^{-1}*A*^{-1}.

### Corollary 1

If *A*_{1}, *A*_{2}, ..., *A*_{r} are *n* x *n* nonsingular matrices, then *A*_{1}*A*_{2}...*A*_{r} is nonsingular an (*A*_{1}*A*_{2}...*A*_{r})^{-1} = *A*_{r}^{-1}*A*_{r-1}^{-1}...*A*_{1}^{-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 *A*^{T} is nonsingular and (*A*^{-1})^{T} = (*A*^{T})^{-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}{ccc|ccc}a&b&c&1&0&0\\d&e&f&0&1&0\\g&h&i&0&0&1\end{array}\right) $

Example