4 Inverse Matrices and Determinants

The Inverse of a Matrix

The inverse (or reciprocal) of a square matrix is denoted by the A-1, and is defined by

A× A−1 = I

For example

	1	1	2		−0.4		0.2	0.6			1	0	0
	1	2	1			−0.2	0.6	−0.2			0	1	0
					×					=
	2	0	1			0.8	−0.4	−0.2			0	0	1

The 2 matrices on the left are inverses of each other, whose product is the unit matrix. Not all matrices have an inverse, and those which don’t are called singular matrices.

Inverting a matrix is a very useful technique, and we will see, but how is it done? Unfortunately it is slightly more complicated than the basic matrix algebra of the previous chapters, so we will need to take a slight detour into the areas of determinants, cofactors and adjoint matrices first.

Determinants

In this section we are simply going to define the determinant and, in later sections, point out some of its properties. A deeper discussion of determinants probably deserves its own paper.

The determinant of a square matrix is a single number calculated by combining all the elements of the matrix. For example, the determinant of a 2 by 2 matrix is

a1,1	a1,2	= a	×a	2,2	−a	2,1	×a
a2,1	a2,2	1,1					1,2

For a 3 by 3 matrix the formula is

a1,1

a1,2

a1,3

a2,2

a2,3

a1,2

a1,3

a1,2

a1,3

a

2,1

a

2,2

a

2,3

= a ×

−a

2,1

×

+a

3,1

×

1,1

a3,2

a3,3

a3,2

a3,3

a2,2

a2,3

a3,1

a3,2

a3,3

The 2 by 2 determinants are called minors. Every element in a determinant has a corresponding minor, formed by deleting the row and column containing that element. For a determinant of order n, the minors are of order (n-1).

In general a determinant of order n is calculated from

∑n (−1)i+1 ×ai,1 ×mi,1 i=1

where m is the minor of a.

Cofactors

The cofactor of an element is the minor multiplied by the appropriate sign

ci,1 = (−1)i+1 ×mi,1

or more generally

ci, j = (−1)i+ j ×mi, j

Adjoint Matrices

Every square matrix has an adjoint matrix, found by taking the matrix of its cofactors, and transposing it, ie if

a1,1

a2,1

A = ..

an,1

a	.	.	a
1,2			1,n
a2,2	.	.	a2,n
.			.
.			.
an,2	.	.	an,n

then the adjoint is