分块对角矩阵和分块次对角矩阵的性质若矩阵P分为3块ABC都可逆分别是234阶方阵,分别在对角、次对角,讨论P的逆矩阵,伴随矩阵,

P=
A 0 0
0 B 0
0 0 C
则 |P|=|A||B||C|,且
P^-1=
A^-1 0 0
0 B^-1 0
0 0 C^-1
P* = |P|P^-1 = |A||B||C|P^-1 =
|A||B||C|A^-1 0 0
0 |A||B||C|B^-1 0
0 0 |A||B||C|C^-1
=
|B||C|A* 0 0
0 |A||C|B* 0
0 0 |A||B|C*
若 P=
0 0 A
0 B 0
C 0 0
则 |P| = (-1)^(2*3*4)|A||B||C| = |A||B||C|.
P^-1 =
0 0 C^-1
0 B^-1 0
A^-1 0 0
P* = |P|P^-1 = |A||B||C|P^-1 =
0 0 |A||B||C|C^-1
0 |A||B||C|B^-1 0
|A||B||C|A^-1 0 0
=
0 0 |A||B|C*
0 |A||C|B* 0
|B||C|A* 0 0