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n A ∈ Mn(K)
rg(A) = n=⇒rgCom(A)=n
rg(A) = n−1 =⇒rgCom(A)= 1
rg(A)≤n−2 =⇒rgCom(A)= 0
.
detCom(A)=det(A)n−1
ComCom(A)
A, B ∈ Mn(C)
A B A
B
B=t(Com A)A∈ Mn(K)
rg A=n−1
C∈ Mn(K)
AC =CA =On
λ∈K
C=λB
A, B ∈ Mn(Z) det Adet B
U, V ∈ Mn(Z)
UA +V B =In
S∈ Sn(R)S

rg(A) = n A
rg(Com(A)) = n
rg(A)≤n−2A n −1
Com(A) = On
rg(Com(A)) = 0
rg(A) = n−1AtCom(A) = det(A).In=On
f g Kn
AtCom(A)
f◦g= 0 Im(g)⊂ker(f) rg(f) = n−1 dim ker(f) = 1
rg(g)≤1
rg(Com(A)) ≤1
rg(A) = n−1n−1
Com(A)6=On
rg(Com(A)) = 1
AtCom(A) = det(A).In
det(A) detCom(A)=det(A)n
det(A)6= 0
detCom(A)=det(A)n−1
det(A) = 0 rg(Com(A)) ≤1< n
det(Com(A)) = 0
rg(A) = n
tCom(Com(A)).Com(A) = det(Com(A)).In= det(A)n−1.In
tCom(Com(A)) = det(A)n−1Com(A)−1
tCom(A).A = det(A).In
tCom(A) = det(A).A−1
t(B)−1= (tB)−1
Com(Com(A)) = det(A)n−2A
rg(A)≤n−1n≥3 rgCom(A)≤1≤n−2
Com(Com(A)) = On
n= 2
A=a b
c d,Com(A) = d−c −b a Com(Com(A)) = A
A B
A B
1
det A
t(Com A)1
det B
t(Com B) = 1
det B
t(Com B)1
det A
t(Com A)
Com(A) Com(B) = Com(B) Com(A)
p
A+1
pInB+1 pIn
Com A+1
pInCom B+1 pIn= Com B+1 pInCom A+1 pIn
p→+∞
Com(A) Com(B) = Com(B) Com(A)

AB =BA = det(A)In
rg A=n A B rg B=n
rg A=n−1 dim ker A= 1 AB =OnIm B⊂ker A
rg B≤1
A n −1
B6=Onrg B= 1
rg A≤n−2A B =On
rg B= 0
rg A=n−1 dim ker A= 1 dim ker tA= 1
X Y
ker A= Vect Xker tA= Vect Y
M∈ Mn(K)AM =MA =On
AM =OnIm M⊂ker A= Vect X
M= (λ1X|. . .|λnX) = XL
L=λ1. . . λn)
MA =OnXLA =OnX6= 0
LA = 0 tAtL= 0 L L =λtY
M
M=λXtY
AM =MA =On
{M∈ Mn(K)|AM =MA =On}= Vect(XtY)
B
C∈ Mn(K)AC =CA =OnB
u, v ∈Zudet A+vdet B= 1 U=ut(Com A)
V=vt(Com B)
(i, j)S
(−1)i+j∆i,j
∆i,j (i, j)S
i j S
S
(i, j) (j, i)
S