This is because
6.The determinant of the product of two square matrices AandBis the product of
determinants, or
IfB=nmmatrix and C=mnmatrix, then
det(In+BC)=det(Im+CB)
If andD=m*mmatrix, then
whereS=D-CA^1 B.
If , then
whereT=A-BD^1 C.
If or then
Rank of Matrix. A matrix Ais said to have rank mif there exists an m*msub-
matrixMofAsuch that the determinant of Mis nonzero and the determinant of every
r*rsubmatrix (where ) of Ais zero.
As an example, consider the following matrix:
A= D
12 34
01 - 10
10 12
11 02
T
rm+ 1
detc
AB
0D
d =detAdetD
detc
A0
CD
d =detAdetD
B= 0 C= 0 ,
detc
AB
CD
d =detDdetT
DZ 0
detc
AB
CD
d =detAdetS
AZ 0
@AB@= @A@@B@
kA= D
ka 11 ka 12 p ka 1 m
ka 21 ka 22 p ka 2 m
oo o
kan 1 kan 2 p kanm
T
Appendix C / Vector-Matrix Algebra 875