考研线性代数最权威老师讲义

···
an
1 a2 · · · an
nΒιβλιοθήκη Baidu1
ai
+
x
a2 + x
···
...
...
...
an
n
1 a2 + x · · ·
... = ( 1 ai + x) ...
...
...
an ...
n 1
ai
+
x
a2
· · · an + x
1 a2 · · · an + x
òIª¡'I(ªn1 Iaii+(2x≤)xin−≤1"n)~I'ai,=C¤enIªÙxn−1, ¦
2 3 1
~ # ¦ " 2: A = 0 2 3, (A∗)−1


©Û:&¦00 2¦_¢"¡,e^_Ý
,u(A∗)−1 ~3: #A´n0"Ý
, ÷vAT = A∗, y²
=
(|A|A−1)−1
=
A 8
.
e y² (1) |A| > 0; (2) n > 2, |A| = 1.
Çؤá§= AB = AC, A = 0,
00 B=C
b. ab = ba,¢´éuÝ
AB = BAؽ¤á§e¤á¡A, B¢"éu'Ý
,¢
Sؤá
éõ'úªØP`{
" 'X(a + b)2 = a2 + 2ab + b2,éuÝ
uC¤
²úªéuÝ
(A+B)2 = A2+AB +BA+B2,
¦5'IªØI IÚI & ' 0'Iª i
j
n−1
.
Mij
. Mij
~5 ¦e¡n0Iª'
1 a1 0 · · · 0
0 1 a2 · · · 0
......................
0 0 0 · · · an−1
an 0 0 · · · 1
Notes1: ÏvIª'Ðm½n±r¦5'IªcIü0O, Xt&O'z! ¡nIª'Ú'"¡, % QOk^\~{¦,C0, , P^Ðm

'Iª I!n^AON Notes1:
. |A + B| =?|A| + |B|, |AB| = |A||B|, |kA| = kn|A|,
´Z%5¿.




1111
3134
~ # ¦ 2
A
=
1
2
0
0
,
B
=
1
2
0
0,
|A|, |B|, |2A|, |A + B|, |2A + B|.
A
0
0
= |A||B|,
A = (−1)m+n|A||B|.
0B
B0
1 1 ··· 1
%%Iªµ d.
x1
x2 · · · xn =
.......................
1≤j<i≤n(xi − xj ).
Notes: %%Ixªn1^−15xOn2−1·I· ·(½xnn−1 )¡1'ê'Iª.
a1 0 0 b1
~ ¦ 10
0 a2 b2 0
0 b3 a3 0
b4 0 0 a4
3
12 3 4
~ ¦ 1 22 32 42
11
1 23 33 43
98 7 6
1Ù Ý
. Ä
1. Ý

: òmnêü¤mI, n§ü>x)Ò§=GX
a11 a12 · · · a1n

a21
'ke¡' S:
(1) AkAl = Ak+l,
(2) (Ak)l = Akl.
4


1111


2311
~ # ¦ 1
A
=
1
2
0
0
,
B
=
1
2
6
2,
2A + B, 3A − 2B, AB, AT , A2B2.
1 0 3 0
1 0 5 3




1004
1232
a22
···
a2n


...
...
...
...



an1 an2 · · · amn
¡mIn
,em = nÝ

. üÝ
'Iê1 ê1u, ¡§´ NÓoFteÝs:
Ý.
XÚtIÙªÓØÓ,þÝ'
£Lþ1L§=aij =1b7ijv, uz¡Ý£
2 ,
−1
ab =
cd
d −b −c a ad − bc
ù«20
kp⬧ïvF4"
5
Notes2:þ¡'úªk­'CGA∗ = |A|A−1"Ï·éu_Ý
'S' Ý
ÙG'õ§éõÿ|^ùúª&òÝ
=¤_Ý
" éuØ# Ø_'ÿ§|^AA∗ = A∗A = |A|E"
1 0 3 0
1 0 3 0




1004
1004
Notes2: Ý
Úê'¦{kü"'O"
½ "¢´éuÝ
a. ab = 0 ⇒ a = 0 b = 0
AB = 0
A = ½0 B = 0"Ò´éuü"
Ý
¦ U´"Ý
"A = B = 0 1 Ò´ù~f"ùÒ`²·ÙG'
... ... .
0 0 · · · an an · · · 0 0
2. Iª'­S.
a. |AT | = |A|.
b. eIª',I£,¤kúÏfk, urúÏfkJ Iª©¡. c. Iª'üI£ü¤p§ÙCÒ.
d. rIª,I£,¤'¤k£'ê \ ,I£,¤'A£þ§¤& '5'Iª'1u¦Iª'.
1Ü© 5êùÂ 1Ù 1ª
. Ä
1 Iª òn2êü¤nI, n§ü>xü^§=GX
a11 a12 · · · a1n a21 a22 · · · a2n
... ... . . . ...
an1 an2 · · · ann
¡n0Iª, Ù¥aij¡Iª'£.
2. Iª'O.
Ý
'=.
AT
=
§ke¡'S (aji)n×m,
:
(1) (AT )T = A, (2) (A + B)T = AT + BT (3) (kA)T = kAT (4) (AB)T = BT AT
Ý
'¦{. ¦k^',
7#vAI=(aijÝ)m
×l,'B=ê(1biju)l×In,uABÝ
a31
~ O 7
|A| =
a21b1 a1b21
b31
a32 a22b2 a2b22 b32
a33 a23b3 a3b23 b33
a34
a24b4 a4b24
.
b34
2
1. .êop.1ª
{: a I½¥'£¡1'§u^%%Iª¦A. b e0QIª'"õĽÂ{§e¢´,I½,0"õ(k")^UI½ UÐmü0?n. c e´G§u^\~{¦,£0§P^UI½UÐm(¢S)½ö&òI ªC¤þn½enIª(S).
0Iª a.
|A| = a11 a21
a12 a22
= a11a22 − a12a21.
'Iª b
a11 a12 · · · a1n
a21 ...
a22 ...
··· ...
a2n ...
=
(−1)τ (j1j2···jn)a1j1 a2j2 · · · anjn
j1j2···jn
an1 an2 · · · ann
(A+B)(A−B) = A2 −AB +BA+B2,
Ásõ\5¿.
4. _Ý
: éuüÝ
÷vAB = BA = E, u¡A´B'_Ý
§±`B´A'_Ý
"¯¢þ§Äk`²AÚBÑ´
"Ý
´kg'§Ø´?ÛÝ
ѱk_Ý
'§_'¡_ Ý
½ÛÉÝ
§Ø_'¡Ø_
½¡ÛÉÝ
:
(1) A+B = B +A, (2) A+(B +C) = (A+B)+C, (3) A+0 = 0+A = A, (4)A+(−A) = 0.
ê¦6½Â. kA = (kaij)m×n,§ke¡'S:
(1) 1A = A, 0A = 0 (2) k(lA) = (kl)A (3) k(A + B) = kA + kB (4) (k + l)A = kA + lA
00014
1111
~8 OIª|A| = 1
2
0
0 .
1030
1004
~9 ¦e¡n0Iª'
a1 + x a2 · · · an
a1 a2 + x · · · an
...
...
...
...
a1
a2 · · · an + x
©ÛµzI'£Ú1§r¤k£Ñ\ I§&
n 1
ai
+
x
a2
AÑ=B1. , Iª´ ê, Iª1 '1=.
2. AÏÝ
µé§þn§en§ü
§é¡é¡Ý
" é¡Ý
:÷vA = AT§êÒ±w{ü'é¡Ý
" é¡Ý
:÷vA = −AT.
3. Ý
'6.
# u Ý
'\{ke¡{ü'S A = (aij)m×n, B = (bij)m×n, A ± B = (aij ± bij)m×n,
­ {.
¥N§o1rteuIs2(òiI¦5¤'b1I,):b2,ª·ê· ·¥{, bIfn'iIªI'|ªÜ.1'b15AIi1­+ªb2A''i2+G. · ·´· +ObnAAiin1§+ÒA1i2 u+ Q· · ·¦+5A'in'Iª§
5 1 23
~ # O ' 3 −1 2 1
Ù¥¥¤τ(jk1j_2 ···'jn)ê_¡ê_. üê.¥eQêQc,¢êQ §ùéꨤ_§ü ~1 τ (436512), τ (n n − 1 n − 2 · · · 2 1) ùĽ¥¡kn!\, O¨'ED, ÏdØ^5¦Iª'§¢´Qe¡ A«G±^: a1. ئIª'§¦Ù¥'.
e. |AB| = |A||B|.
ؽ¤á Notes: |A + B| = |A| + |B|
.
2 1 41
~4 O 3 −1 2 1 '. 1 2 32
5 0 62
3. Iª'Ðm½n.
ê{fª§ {fª ½ |A| = ai1Ai1 + ai2Ai2 + · · · + ainAin, Aij = (−1)i+j, Aij
x−2 x−1 x−2 x−3
~ OIª 2x − 2 2x − 1 2x − 2 2x − 3
6
|A| =
.
3x − 3 3x − 2 4x − 5 3x − 5
4x 4x − 3 5x − 7 4x − 3
43000
~ OIª 1 4 3 0 0
7
|A| = 0 1 4 3 0 .
00143
Notes3: Ý
'AS.|^Ý
'Oúª, ±&
a1 |A∗| = |A|n−1
e _ u a3 A , (A−1)∗ = (A∗)−1 c a5 (A∗)∗ = |A|n−2A(98 )
= '(ciIj)mê×.Ýn
, u'c¦ij{=kelk=¡1'aikSbkl. :Ý

(1) (AB)C = A(BC),
(2)A(B + C) = AB + AC, C(A + B) = CA + CB
(3)k(AB) = (kA)B = A(kB).
Ý
'. #A´n0
, uA±g¦, kA¦Ak, ¡A'kg. Ý
=
(A−1)k, (Ak)T
=
(AT )k.
5.Ý
µA∗'A'ê{fª|¤Ý
'=(½=BéN´)"A∗'­
NoStes1:´þA¡A∗=ÑA'∗A_=Ý|
A|úE,ªeÝk
nAØ_d,u,ØÏ
v20Ý
©±¦Ø_¬Ý^
Ù,OA−1 .=¨|AAn∗|
. =
x 2 3 4−x
~ # 3 −2x 5 2 f (x) =
1 , ¦x3Úx4'Xêõ¨.
23x 6
1 5 0 2x
a2. "éõ'Iª§ù$'zké¨'ê\.
a1 0 · · · 0 0 · · · 0 a1
~ ¦ 0 a2 · · · 0 0 · · · a2 0
3 ... ... . . . ... , ...
6 |A| =
,
1 2 32
M41 + M42 + M43 + M44 .
5 0 62
4. ­'AIª.
a b
þnIª: enIª:
Iª¥Qée'£0, Iª¥Qéþ'£0,
|A| |A|
= =
a11a22 a11a22
· ·
· · ann. · · ann.
OéIªµ c.
"éu
k
_ A ⇔ |A| = 0
Notes1: éu
, `A´B'_Ý
AB = ½E BA = E=,Ù¥¤áu, g,¤áØs¨y. Notes2: _Ý
'S. eA, B_, u
(AB)−1
=
B−1A−1,
(cA)−1
=
1 c
A−1,
(AT
)−1
=
(A−1)T , (Ak)−1