Matrix division is not defined, but provided a square matrix A has a non-zero associated determinant jA j, a multiplicative inverse A1 exists with the property that


Thus A1 and A are mutually inverse, in the sense that A1 is the multiplicative inverse of A , and A is the multiplicative inverse of A1. To discover how to compute A1 from A , we need to make use of the Laplace expansion rule for determinants and a result mentioned only in Problem 25 of Problems

52 that the sum of the products of the elements in any row (or column) of a determinant with the corresponding cofactors of a different row (or column)

is zero. It then follows that if C is the matrix of cofactors of A ,


½ A ½ 0 0 0 0 ½ A ½ 0 0 0 0 ½ A ½ 0 0 0 0 ½ A ½

2 66664

3 77775 ½ A ½ I ,

and so, provided j A j"/0,



½ A ½


½ A ½

AI :

Thus, the inverse A1 of A is given by


½ A ½ , provided ½ A ½"0:

The transpose of the matrix of cofactors CT is important in its own right and it is called the adjoint matrix and written adj A . Thus, provided jA j"/0,

A1 adj A

½ A ½ :

A square matrix A is said to be non-singular if j A j"/0 and is said to be singular if j A j/0. Thus only non-singular square matrices possess inverses

where 0 denotes the null matrix.