The Mathematics of Financial Modelingand Investment Management

5-Matrix Algebra Page 159 Wednesday, February 4, 2004 12:49 PM

Matrix Algebra 159

only if the number of columns (i.e., the number of elements in each row)

of the first matrix equals the number of rows (i.e., the number of ele-

ments in each column) of the second matrix.

The following two distributive properties hold:

CA ( + B) = CA + CB

(AB+ )C = AC + BC

The associative property also holds:

(AB)C = A BC ( )

However, the matrix product operation is not commutative. In fact, if A

and B are two square matrices, in general AB ≠BA. Also AB = 0 does

not imply A = 0 or B = 0.

Inverse and Adjoint

Consider two square matrices of order n, A and B. If AB = BA = I, then

the matrix B is called the inverse of A and is denoted as A–1. It can be

demonstrated that the two following properties hold:

■ Property 1. A square matrix A admits an inverse A–1 if and only if it is

nonsingular, i.e., if and only if its determinant is different from zero.

Otherwise stated, a matrix A admits an inverse if and only if it is of full

rank.

■ Property 2. The inverse of a square matrix, if it exists, is unique. This

property is a consequence of the property that, if A is nonsingular, then

AB = AC implies B = C.

Consider now a square matrix of order nA = {aij} and consider its

cofactors αij. Recall that the cofactors αij are the signed minors

(– 1 )(ij+)M of the matrix A. The adjoint of the matrix A, denoted as

ij

Adj(A), is the following matrix:

α 11 , · α 1 ,j · α 1 ,n , ,

T α

11 · α 21 · αn, 1

· ··· · ·····

Adj ()A = αi, 1 · αij, · αin, = α 1 ,i · α 2 ,i · αni,

· ··· · ·····

αn, 1 · αnj, · αnn, α 1 ,n · α 2 ,n · αnn,