4 Inverse of a matrix

In ordinary algebra the division operation $a \div b$ is equivalent to multiplication of the reciprocal $a \times \frac{1}{b}$. Thus one binary operation, division, has been replaced by two operations, one binary (multiplication) and one unary (forming $\frac{1}{b}$). In matrix algebra we make an equivalent substitution of operations, and we call the unary operation inversion. We write the inverse of the matrix $\bf A$ as ${\bf A}^{-1}$, and calculate it so that

$${\bf AA}^{-1} = {\bf I}$$

and

$${\bf A}^{-1}{\bf A} = {\bf I} \; ,$$

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 $3 \times 3$ matrix in order to fully understand the concept of a matrix inverse.