Page 446 - Schaum's Outline of Theory and Problems of Signals and Systems

REVIEW OF MATRIX THEORY

B. Rank of a Matrix:

The number of linearly independent column vectors in a matrix A is called the column
rank of A, and the number of linearly independent row vectors in a matrix A is called the
row rank of A. It can be shown that
Rank of A = column rank of A = row rank of A (A.19)

Note:
If the rank of an N x N matrix A is N, then A is invertible and A-' exists.

A.4 DETERMINANTS

A. Definitions:
Let A = [aij] be a square matrix of order N. We associate with A a certain number
called its determinant, denoted by detA or IAl. Let M,, be the square matrix of order
(N - 1) obtained from A by deleting the ith row and jth column. The number A,j defined
by

is called the cofactor of a,,. Then det A is obtained by

N
detA=IAl=CaikAik i=1,2, ..., N (A.21~)
k=l
N
or detA = (A1 = C akjAk, j = 192?. . . N ( A.21b)
k=l

Equation (A.21~) is known as the Laplace expansion of IAl along the ith row, and Eq.
(A.21b) the Laplace expansion of IAl along the jth column.

EXAMPLE A.10 For a 1 x 1 matrix,

A= [a,,] --, IAl =a,,
For a 2 x 2 matrix,

For a 3 x 3 matrix,