Page 447 - Schaum's Outline of Theory and Problems of Signals and Systems

REVIEW OF MATRIX THEORY [APP. A

Using Eqs. (A.21~) and (A.231, we obtain

B. Determinant Rank of a Matrix:
The determinant rank of a matrix A is defined as the order of the largest square
submatrix M of A such that det M # 0. It can be shown that the rank of A is equal to the
determinant rank of A.

EXAMPLE A.ll Let

Note that IAl = 0. One of the largest submatrices whose determinant is not equal to zero is

Hence the rank of the matrix A is 2. (See Example A.9.)

C. Inverse of a Matrix:
Using determinants, the inverse of an N x N matrix A can be computed as
1
~ - 1 ~ - adj A
det A
...
A11 A21 ANI
...
... A:]
AIN
and adjA=[~,,]~=[~: A: ...
ANN

where A,, is the cofactor of aij defined in Eq. (A.20) and "adj" stands for the adjugate (or
adjoint). Formula (A.25) is used mainly for N = 2 and N = 3.

EXAMPLEA.12 Let