Revision as of 07:30, 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 = I_n. We call B the inverse of A and denote it as A^-1. Thus, AA^-1 = A^-1A = I_n. In this case, A is also called nonsingular.

Example.

$ A = \left(\begin{array}{cc}4&3\\3&2\end{array}\right) $
$ A^{-1} = \left(\begin{array}{cc}-2&3\\3&-4\end{array}\right) $

$ AA^{-1} = \left(\begin{array}{cc}4&3\\3&2\end{array}\right) $$ \left(\begin{array}{cc}-2&3\\3&-4\end{array}\right) = $ $ \left(\begin{array}{cc}1&0\\0&1\end{array}\right) $

and
$ A^{-1} = \left(\begin{array}{cc}-2&3\\3&-4\end{array}\right) $$ \left(\begin{array}{cc}4&3\\3&2\end{array}\right) = $$ \left(\begin{array}{cc}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, ..., A_r are n x n nonsingular matrices, then A₁A₂...A_r is nonsingular an (A₁A₂...A_r)^-1 = A_r^-1A_r-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 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}{cc}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}{cc}d&-b\\-c&a\end{array}\right) = \frac{1}{ad - bc}\left(\begin{array}{cc}d&-b\\-c&a\end{array}\right) $

Example

$ A = \left(\begin{array}{cc}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}{cc}4&-2\\-3&1\end{array}\right) $
$ A^{-1} = \left(\begin{array}{cc}-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. Find A^-1 of
$ A = \left(\begin{array}{cc}1&2\\3&4\end{array}\right) $