ABSTRACT
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
A1AAA1I :
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 ,
ACTCTA
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,
A
CT
½ A ½
CT
½ A ½
AI :
Thus, the inverse A1 of A is given by
A1 CT
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.