Anon

Fundamentals of Matrix Algebra 389

where the sum is extended over all permutations (j 1 ,... , jn) of the set (1,2,... ,
n) and t(j 1 ,... , jn) is the number of transpositions (or inversions of positions)
required to go from (1,2,... ,n) to (j 1 ,... , jn). Otherwise stated, a determi-
nant is the sum of all products formed taking exactly one element from each
row with each product multiplied by ()− 1 tj(,...,^1 jn). Consider, for instance, the
case n = 2, where there is only one possible transposition: 1, 22 ⇒ ,1. The
determinant of a 2 × 2 matrix is therefore computed as follows:

A=−()() 11 aa+− aa=−aa aa

0

11 22

1

12 21 11 22 12 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+ Mij is called the cofactor of aij and is gener-
ally denoted as αij.
A square matrix A is said to be singular if its determinant is equal to
zero. A 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 nonsingular
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 simulta-
neous equations of the following form:

++ =

++ =

ax ax b

ax ax b

...

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

...

mm

nmmm

11 11 1

11 1

The n × m matrix:

=

⋅⋅

⋅⋅⋅⋅⋅

⋅⋅

⋅⋅⋅⋅⋅

⋅⋅





      





      

aaa

aaa

aaa

A

jm

iijim

nnjnm

1,11,1,

,1 ,,

,1 ,,

formed with the coefficients of the variables is called the coefficient matrix.

The terms bi are called the constant terms. The augmented matrix (^) []Ab—