The Mathematics of Financial Modelingand Investment Management

5-Matrix Algebra Page 149 Wednesday, February 4, 2004 12:49 PM

Matrix Algebra 149

Consider, for instance, the case n = 2, where there is only one possi-

ble transposition: 1,2 ⇒ 2,1. The determinant of a 2×2 matrix is there-

fore computed as follows:

A = (– 1 )^0 a^1

11 a 22 + (–^1 )a 12 a 21 = a 11 a 22 – a 12 a 21

Consider a square matrix A of order n. Consider the matrix Mij

obtained by removing the ith row and the jth column. The matrix Mij is

a square matrix of order (n – 1). The determinant Mij of the matrix

Mij is called the minor of aij. The signed minor

(– 1 )(ij+) M

ij

is called the cofactor of aij and is generally denoted as αij. The r-minors

of the n×m rectangular matrix A are the determinants of the matrices

formed by the elements at the crossing of r different rows and r different

columns of A.

A square matrix A is called singular if its determinant is equal to

zero. An n×m matrix A is of rank r if at least one of its (square) r-minors

is different from zero while all (r + 1)-minors, if any, are zero. A non-

singular square matrix is said to be of full rank if its rank r is equal to

its order n.

SYSTEMS OF LINEAR EQUATIONS

A system of n linear equations in m unknown variables is a set of n

simultaneous equations of the following form:

a 11 , x 1 + ... + a 1 , mxm = b 1

........................

an, 1 x 1 + ... + a 1 , mxm = bm

The n×m matrix

a 11 , ·a 1 , j· a 1 , m

· ··· ·

A = ai, 1 · aij, · aim,

· ··· ·

an, 1 ·anj, · anm,