Théorème de Desnanot-Jacobi

M n ×n Mj
iM i
j|M|= det(M)
|M|×|M1,n
1,n |=|M1
1|×|Mn n| − |M1 n|×|Mn 1|
M n ×ntcom(M)i.e.
(i, j)Mi
j
tcom(M) = 
|M1
1| −|M2 1| · · · (−1)n+1|Mn 1|
−|M1
2| |M2 2| · · · (−1)n+2|Mn 2|
(−1)n+1|M1
n|(−1)n+2|M2 n| · · · |Mn n|
i M i
(−1)i+1 ¡mi1|Mi
1| − mi2|Mi 2|+. . . + (−1)n−1min|Mi n|¢=|M|
j M i tcom(M)j6=i
i j
mj1|Mi
1| − mj2|Mi 2|+. . . + (−1)n−1mjn|Mi n|= 0
M.tcom(M) = 
|M|0 0 · · · 0
0|M|0· · · 0
0 0 0 · · · |M|
|M|.|tcom(M)|=
|M|nn2m1,1, . . . , mn,n
|M|
|tcom(M)|=|M|n−1
M∗tcom(M) (i, j) 2 ≤k≤n−1 0
i6=j1i=j
M∗=
|M1
1|00· · · 0 (−1)n+1|Mn 1|
−|M1
2|10· · · 0 (−1)n+2|Mn 2|
|M1
3|01· · · 0 (−1)n+3|Mn 3|
(−1)n|M1
n−1|00· · · 1−|Mn n−1|
(−1)n+1|M1
n|00· · · 0|Mn n|
M M∗
M.M∗=
|M|m12 m13 ··· 0
0m22 m23 ··· 0
0m32 m33 ··· 0
0mn2mn3· · · |M|
|M|2|M1n
1n|M∗|M1 1||Mn
n|−|M1 n||Mn
1|
|M|¡|M1
1||Mn n|−|M1 n||Mn 1|¢=|M|2|M1n 1n|
m1,1, . . . , mn,n
|M|

M n ×n M = (ai,j )
(A, B)A=M B (n−1) ×(n−1) 1
B A0(n−1)×(n−1) B0(n−2)×(n−2)
a0
i,j =ai,j ai+1,j+1 −ai,j+1 ai+1,j
bi,j
, b0
i,j =ai+1,j+1
A02×2A
B B0A
•B
•A B A
A=
1−2−1 3
2 1 −1 2
−1−2 1 −3
0−1−1 2
B=
111
111
111
A0=
531
−3−1 1
1 3 −1
B0=µ1−1
−2 1 ¶
A00 =µ4−4
4−2¶B00 = (−1)
A000 = (−8) B000 = ( )