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ER
0
A B M3(R)A
B P ∈GL3(R)
B=P−1AP
A∼B A B
∼ M3(R)
u E i j
wker ui+jE
w(x) = uj(x)
Im w⊂ker ui
dim(ker ui+j)≤dim(ker ui) + dim(ker uj)
u E u3= 0 rg u= 2
dim(ker u2)=2
a u3(a)6= 0 (u2(a), u(a), a)
E
U u V v =u2−u
AM3(R)
T=
1α β
0 1 γ
0 0 1 
A A−1
N=
0α β
0 0 γ
0 0 0 
P∈GL3(R)
P−1AP =T=I3+N
A
N3
P−1A−1P=I3−N+N2
N= 0 A A−1
rg(N)=2 M=N2−N
N
010
001
000
M
M3rg(M)
M N
A A−1
rg(N)=1 M=N2−N
A A−1
A=
1 0 0
0 0 −1
0 1 2 
(a, b, c)E u E A
ker(u−IdE)E
(e1, e2)

(e1, e2, c)E u
A A−1
T=
1α β
0 1 γ
0 0 1 
M3(R)
T
∼
A∼A P =I
∼B P
B=P−1AP ⇒A=Q−1BQ
Q=P−1B∼A
A∼B B ∼C P Q
B=P−1AP
C=Q−1BQ)⇒C= (Q)−1(P)−1A(P Q)=(P Q)−1A(P Q)
C∼A P Q
B=P−1AP ⇒det(B) = det(P−1AP ) = det(P−1) det(A) det(P)
= det(A) det(P) det(P−1) = det(A) det(P P −1) = det(A)
x∈Im w y ∈ker(ui+j)
x=w(y) = uj(y)⇒ui(x) = ui+j(y)=0E
x∈ker ui
Im w⊂ker ui
w
dim(ker ui+j) = dim(ker w) + dim(Im w)
ker w= ker ui+j∩ker uj= ker uj
Im w⊂ker ui
dim(ker ui+j)‘ dim(ker ui) + dim(ker uj)

u u3=O E
dim(ker u2)≤2 dim(ker u)=2
3 = dim(ker u3)≤dim(ker u2) + dim(ker u)≤dim(ker u2)+1
dim(ker u2)=2
dim(ker u2) = 2 u2
a u2(a)6= 0Eu(a)a
(u2(a), u(a), a)
u2
a u
u(a)
v=u2−u U u V v
(u2(a), u(a), a)
U=
010
001
000
, U2=
001
000
000
, V =
0−1 1
0 0 −1
0 0 0 
det A= det T= 1 T
N3
(I3+N)(I3−N+N2) = I3−N3=I3
(P−1AP )−1=I3−N+N2
(P−1AP )−1=P−1A−1P
N= 0 A I I P A
I A A−1
I
rg(A) = 2 M=N2−N N3= 0
N
010
001
000
M
−
010
001
000
+
001
000
000
=
0−1 1
0 0 −1
0 0 0 
M3= 0 M
010
001
000
,
0−1 1
0 0 −1
0 0 0 
M
M N I +M I +N
A∼I+N
A−1∼(I+N)−1=I−N+N2=I+M∼A
rg(A)=1 M=N2−N n
N n3= 0
Im n⊂ker n
Im nker n
a n(a) = λa λ
n3(a) = λ3a6= 0

Im n⊂ker n
(a, b, c) (c) Im n(a, b) ker n
n
001
000
000
N2= 0 M2=−N
(I+N)−1=I−N
A∼I+N A−1∼I−N
I+N I −N
N n (a, b, c)−N n (−a, b, c)
A∼A−1
u−IdE(a, b, c)
A=
0 0 0
0−1−1
0 1 1 
dim(ker(u−IdE)) = 2
(e1, e2) = (a, b −c)
ker(u−IdE) (b−c) Im(u−IdE)
(a, b −c, c)E
(a, b, c) (a, b −c, c)
a=a, b = (b−c) + c, c =c
u(c) = −b+ 2c=−(b−c) + c u (a, b −c, c)
1 0 0
0 1 −1
0 0 1 
A A−1
A I +N N
T
A=
1 0 0
0 2 0
0 0 1
2
= Mat
(a,b,c)u
A−1=
1 0 0
01
20
0 0 2
= Mat
(a,b,c)u−1= Mat (a,c,b)u
A
T
tr A=1+2+ 1
26= tr T= 3