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4 Inverse of a matrix

In ordinary algebra the division operation is equivalent to
multiplication of the reciprocal
. Thus one
binary operation, division, has been replaced by two operations, one
binary (multiplication) and one unary (forming ). In
matrix algebra we make an equivalent substitution of operations, and
we call the unary operation *inversion.* We write the inverse of
the matrix as , and calculate it so that

and

where **I** is the identity matrix. In general the inverse of a
matrix is not simply formed by finding the reciprocal of each element
(this holds only for scalars and diagonal matrices^{}), but is a more complicated operation involving the
determinant.
There are many computer programs available for inverting matrices. Some
routines are general, but there are often faster routines available if the
program is given some information about the matrix, for example, whether it is
symmetric, positive definite, triangular, or diagonal. Here we describe one
general method that is useful for matrix inversion; we recommend
undertaking this hand calculation at least once
for at least a matrix in order to fully understand the
concept of a matrix inverse.

**Subsections**

** Next:** 1 Procedure:
** Up:** 2 Unary Operations
** Previous:** 3 Trace of a
** Index**
Jeff Lessem
2002-03-21