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1 A n n B AB = BA = I = n B A (inverse matrix) A -1 A A (determinant) det A (1) det A 6=0 ⇐⇒ A A -1 A~x = ~ b 1.1 1 (p) (q ) 2 :1 i, j n, i 6= j P λ ij : i λ j Q ij : i j 1–1

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Page 1: A n B n B A A A~x - Ehime Universityrims-sym.math.sci.ehime-u.ac.jp/users/tsuchiya/math/... · 2005-01-13 · E 7DF yQ NTudb i $ % @ 1 G[ 2b) c T 0 HB # # .aL8 ! " $ %. i 3 P

1� � � � � � ��� �

A ��� �������������������������� n �!#"�$&%('*)+��,.-0/1����� n �!�"�$% B 243�5(/1�

AB = BA = I = n &6�7�$&%'98���'�:�� B A ��;#<&= (inverse matrix) '?>�@&� A−1 '1AB�&C�DE �GF0H�IKJLC�� E 'M�N�G!O"P$#% A Q9RS/UTO� A ��<#=BV (determinant) 'U>BW�X��Y det A 243P5(/1�(1) det A 6= 0 ⇐⇒ A �4Z�$&% A−1 243P5�)+�2�[\J^]O_`'1�(a E '�b�O8��Mc`dfe�[hg�Q4i “ ��jPk#lhg�Q ” m#) E '?H��B�?D “ �Pjk&l�g ” '?�naL�&�Oi o&p

"#qBr A~x = ~b ?k(d's�naf]NtuQN]�vOTOwyxB�0's�na{z0|&Hh���MD�$B%Br���}�~y�b�B���P��Jf�N��g&H��8P�B2���$�%��b$&%�r�2B8��h�����+aUQ�}�~���X+�����&2&�#���B��a1Q4�N�B�uG���)+� E '�u���+Q�)+�?D

1.1 �K�\�������n������GTPi� �¡N¢.�u'�:M£h] 1 �"Pq�rh��k�¤�H�iy¥§¦�¨�¤&©ª'1�naM-b���¡O«S/sCbD�¦

¨#¤0�b¬O­n'98h�N�&��i(p) ���9r.�}N�#®K/?T�i�¯��Nr�Qs°±x&�(q) 2 _���r.M²BX#³+x&�

's�na E '9H��PvPC4D E Xu9$B%��N´Pµ�H4´h��¶+x��0'siN ���$B%��N<�·O¸&¹hº�»�Q8h� : 1 ≤ i, j ≤ n, i 6= j '&/sC�'�:���

��

¬O¼B½p

P λij: ¾ i $y λ ®n/UTO�4¾ j $yQ?°±x#���

¬P¼B½p

Qij: ¾ i $n'*¾ j $+�¿P¶&)+�sD1–1

Page 2: A n B n B A A A~x - Ehime Universityrims-sym.math.sci.ehime-u.ac.jp/users/tsuchiya/math/... · 2005-01-13 · E 7DF yQ NTudb i $ % @ 1 G[ 2b) c T 0 HB # # .aL8 ! " $ %. i 3 P

� xNWN�G$#% A � (1, 1)- [�� a11 2 0 H&��8N�ª'#/���¬O¼B½p

P−a21/a11

12 = ¥ ¾ 1 $×(−a21/a11) + ¾ 2 $h©�� �$B)��0'U�

(a11 ×

(−a21

a11

)a12 ×

(−a21

a11

)· · · a1n ×

(−a21

a11

))

+ (a21 a22 · · · a2n)

(0 −a21a12

a11

+ a22 · · · −a21a1n

a11

+ a2n)

's8�J{�����BQ (2, 1)- [�� a21 24¦O¨+�LXy�UD���� ��Qb� k = 3, · · · , n Q9R`/1T���¬�¼½p

P−ak1/a11

1k ���J��S/����B)u��'1� A ��¾ 1 %0� a11 ���&��[�0�����&¦�¨±�*Xu�?D�#���N� a11 6= 0 '��na���}��0-�'?H#���#��H��B�?Du-h/�� a11 = 0 8��sWN� a21, a31,

· · · , an1 �� ���� 0 H&8��.-��� �#)bD�)#c�TP2 0 8��?WN� �! A ��¾ 1 %��4�#"�$&%')( '1�na E 'GQ�8��?Du-h/�� a21, a31, · · · , an1 �� �H 0 H&8N�.-b�P2��GX�WN���B�� �*�+��,P8�-�� - � x�W��/.�RPj�21032�QP80�B-b�54�7698PD��OX02 aj1 H����0'*)X�WO��¾ 1 $('*¾ j $u�¬�¼B½

pQ1j H�¿P¶#)+�sD&)+��'U� a11 6= 0 '98h�N��HP���BH

I�vPC��ua1Q4��¬#¼B½p

P λ2

12 , · · · , P λn

1n � J:�S/+�;��)�� E '�Qh�yJ{� a21, a31, · · · ,

an1 2O¦�¨�Hh:B�sD�_h��J{i4$��4¬�¼B½p

P λij, Qij ªaL�ud $&% A Q�<�=`�?>��0'?i

A :=

a11 a12 . . . a1n

a21 a22 . . . a2n

......

. . ....

an1 an2 . . . ann

=⇒

a′11 a′

12 . . . a′1n

0 a′22 . . . a′

2n...

.... . .

...

0 a′n2 . . . a′

nn

���+a�Q�½P¶0Hh:�� E 's2&�����sD�@�#i�¾ 2 % �A�@BhQ0- E ��½P¶+C� J:�B) E 'bQ��uJ{i� 0�ED�F�2NmS��X�C E 'QP8��?DG�H

1 )&chT#� n #!P"�$B% A Q�¬�¼&½p

P λij, Qij 7�JIyQ�<J=��K>��0'?i

A :=

a11 a12 . . . a1n

a21 a22 . . . a2n

......

. . ....

an1 an2 . . . ann

=⇒

a′11 a′

12 . . . a′1n

0 a′22 . . . a′

2n...

. . .. . .

...

0 . . . 0 a′nn

'U�napQb½P¶0Hh:B�MD

1–2

Page 3: A n B n B A A A~x - Ehime Universityrims-sym.math.sci.ehime-u.ac.jp/users/tsuchiya/math/... · 2005-01-13 · E 7DF yQ NTudb i $ % @ 1 G[ 2b) c T 0 HB # # .aL8 ! " $ %. i 3 P

E �7D�FyQ��NTudb��i�$�%����@���1���G[���2b)�c�T 0 HB�#�#�.aL8�!�"�$�%.�i��3 �P<B= (upper triangular matrix) '?�Ka D��#C��&Q�i� .x&� X0C�!�"�$�% A ��3 �O$&%yQb½P¶&)+��C��P��� (������� 4i�������� "!��G´�#�$yQ&%�'(/UT�( E a DE ��� (����)�� 4i+*-,�.�/ (forward elimination) '?>�8 E 'M2��B�sD0�1325476�8

2 ( *�,�.�/ )

²�9 : 2 �&���;:&� n ' n × n !P"#$&% A := (aij)i,j=1,··· ,n.

for i = 1, 2, · · · , n − 1 do {

|aqi| = max{|aji| : i ≤ j ≤ n} '98h� q ��&) ;

if |aqi| = 0 then continue;

else

¾ i $n'*¾ q $��¿O¶ ;

for k = i + 1, · · · , n do {

¾ i $� −aki/aii ®K/sT4¾ k $uQs°±x&� ;

}

}

< 8y��=B�OiP�h��� (������� NC?>GvOT�iO @0Q A 2O� 3 �O$�%yQ�½P¶��1X+� E '4iE���A��T+dCB+�s�4DE&E H�i�$���¬#¼B½

pP λ

ij �Oi i < j ��D#i� ��p��$&%

1. . .

1...

. . .

λ . . . 1. . .

1

,

R���[�����)&chT 1

(j, i)- [��� λ

��X���E��+F#R��O[�����)&chT 0

1–3

Page 4: A n B n B A A A~x - Ehime Universityrims-sym.math.sci.ehime-u.ac.jp/users/tsuchiya/math/... · 2005-01-13 · E 7DF yQ NTudb i $ % @ 1 G[ 2b) c T 0 HB # # .aL8 ! " $ %. i 3 P

4i4$#% A QC��� �?����� E 'sH� ���Hh:B� E '�Q��Pz#)+�?D� �@#i�$&%&����P���� ��}O~u���B`/Ui� � � O$B)BX�WOiN) �+�.����� E '92#H�:��sD3�9�&i�$��4¬¼B½p

P λij 'U����$&%u;����n/Ui E �b$&%u P λ

ij '��B) E '�Q�)��sD7�uQNi�$���¬#¼B½

pQij �Oi� ���$#%

Qij :=

1. . .

0 . . . 1...

. . ....

1 . . . 0. . .

1

,

(i, i)- [�� , (j, j)- [���� 0

��X���E���R��O[����b)&chT 1

(j, i)- [�� , (i, j)- [��� 1

�OX���E�� F&R��O[����b)BchT 0

A QC��� �?���B� E '�Qh�.J^ ����*X+� E '�����A��� E 'M2#Hh:B�sD E X �?��$%u��yvPTN �@hQ� � /?T < ��'?i

P λijP

−λij = P−λ

ij P λij = I, QijQij = I

2�[\Jf]P_ E 'M2�)��9�P���sD E�E H�iG$#% I ��i n #6�7&$#%�H��B�sD�_��`J i P λij,

Qij �b!�� (regular) H���J�iO�N_ (P λij)

−1 = P−λij , Q−1

ij = Qij 2�[\Jf]P_`'1�(a E '?2����vOCbD3�&��� E '���+' �OT DJF 1 �A�:9¶�x&��'?i� �� DJF����� :

G�H3 ��z0� n #!P"#$&% A Q4i�$���¬#¼B½

p��$B% P λ

ij, Qij 7�JIuQ�����T��d�'Ui A �� 3 ��$#%�Q�½O¶hH�:#�?D�_��±J�i9!���8G$�%0� P1, P2, · · · , PM a �.d�6�8ª'Ui PMPM−1 · · ·P2P1A 2�� 3 �O$�%S'M8��&�yaLQG)u� E '?2PH�:&�UDE#E H�i�� Pk ��i P λij � Qij �P���#Xh�#H����sD

1.2 � ���� "! A~x = ~b #%$ &�MTPi7D�F 3 �i

o#p"PqBr A~x = ~b �9k�¤�Q ��=�)u� E '�GwyxB�ya D1D;F 3 Q��

T('+� PMPM−1 · · ·P2P1 ��+' �NT R '�A`d E '�Q./4�ua D�_���J�iR := PMPM−1 · · ·P2P1

1–4

Page 5: A n B n B A A A~x - Ehime Universityrims-sym.math.sci.ehime-u.ac.jp/users/tsuchiya/math/... · 2005-01-13 · E 7DF yQ NTudb i $ % @ 1 G[ 2b) c T 0 HB # # .aL8 ! " $ %. i 3 P

'1}O~#)+�sD�� Pl ��! ��$&% B�vOCB��HPi R -s! ��$&%0Hh���sD� �@�iR−1 = P−1

1 · · ·P−1M

H��B�MD E&E H�i R o#p"#qBr A~x = ~b � ��� Q �����s���4C�"Pq�r RA~x = R~b

w+x��ua D#$B% R ��!���B�vPCB��H&i���_0��"�qhr A~x = ~b, RA~x = R~b ���j�H���9DB_���J{i ~x ∈ R

n 2 RA~x = R~b ��k+8��MW�i ~x � A~x = ~b ��k.Hh�±J�i����NZª--���+=4[�JL]#_PDª�4vPT#i A~x = ~b 9k(d9C �&Q���i RA~x = R~b sk �PW�����D EE H#i U := RA, ~c := R~b '�(±d D3���hQ < ���ua�Q - ���b���Oi��~u PQ�m(/?C��yaQ�4 i��0CB8�"�q�r U~x = ~c �� h60QMk �&�sD� �Pif$�% U = RA �Ni*� 3 ��$�%AB�v�C E 'G����B4��a D&�?v�TOif"�q#r U~x = ~c

�OiO h���ua*8p./UTP��� :

U~x = RA~x =

u11 u12 . . . u1n

0 u22 . . . u2n

.... . .

. . ....

0 . . . 0 unn

x1

x2

...

xn

=

c1

c2

...

cn

= ~c = R~b.

-�/?i U �bR��P[�� ujj, j = 1, · · · , n 2N)&chT 0 HB8���8��sWOiO_h��J i(2) u11 6= 0, u22 6= 0, · · · , unn 6= 0

8 �9W�i E �b"�q�r0� xn � �����uQ� h6�QMk �B�?D�_���J�iunnxn = cn =⇒ xn =

cn

unn

un−1,n−1xn−1 + un−1,nxn = cn−1 =⇒ xn−1 =1

un−1,n−1(−un−1,nxn + cn−1)

...

'U�Ka9�ua�QNi��B� ���uQMk��BTP� �OW��9����H����MD E ������:O�Oi ��z0� ~c ∈ Rn

QGR�/1T�����H&�&� E '� ��z�).�UD E �����u:P�i�������� (backward insertion) '>/8 E 'U2&�&�1D"� � ��� &! �M´�#�$�QC� �!N²0���"�y:OGA�dP'1iL ����.a Q�8�� :

1–5

Page 6: A n B n B A A A~x - Ehime Universityrims-sym.math.sci.ehime-u.ac.jp/users/tsuchiya/math/... · 2005-01-13 · E 7DF yQ NTudb i $ % @ 1 G[ 2b) c T 0 HB # # .aL8 ! " $ %. i 3 P

0�1325476�84 ( ����� � )

²�9 : � 3 ��$&% U ' n � $�% ' ( ~c ∈ Rn. C-Bª/?i U ��R��P[��0�b)&c�T

0 HB8N�±'+�h}P)+� :

xn =cn

unn

;

for i = n − 1, n − 2, · · · , 1 do {

xi =1

uii(ci − ui,i+1xi+1 − · · · − uinxn);

}

�B�BH�i A �½�¶�/?C U Q�R�/UT�iU 2���� (2) ��.C1>yWOi ��z0� ~c ∈ R

n Q�_���T�"&q�r U~x = ~c - 'A~x = ~b 4��Gk ~x ∈ R

n ���_E '�2h�B��vOCOD E ��D�i ~x ·��� ��±-s[ JU]B_ E '�Q��Bz\/P�±a D�_��(J�i ~x1,

~x2 ∈ Rn 2 U~x = ~c �bk�H�����8��MWOi U ~x1 = ~c, U ~x2 = ~c 8#��H&i 2 _��4r����y.'

�0' U( ~x1 − ~x2) = ~0 �+�sD E#E H#i���� (2) ��-h'MH(3) -0/ U~y = ~0 8��9WOi ~y = ~0 H����2b[�Jf]P_ E '�Gm#) E 's2#H�:��4��H�i ~x1 = ~x2 H���� E 's2E #��Hh:��sD3�#��H�i �� D;F�2���y�1X�CbDG�H

5 A 9½P¶�/UC U �bR-�P[� u11, u22, · · · , unn 2�)#c�T 0 H&8 �9X�W�i �z0� ~c ∈ R

n Qb_B�BT�"Pq�r U~x = ~c ' A~x = ~b �Oi �PzOk ~x ∈ Rn ���_OD

����� H1. �h� (3) Mm�>#D ������ H2. A s½O¶S/1C U ��R���[�� u11, u22, · · · , unn �B bH 0 ��-���2B�GX�WNi

op"Nq#r U~x = ~c 2Mky���C�84�0��a^8 ~c 2�345#).� E 'GUm�>OD E � DOi�-���� = ~b =

R−1~c Q9R`/1T#��i A~x = ~b �sku��hC�8���D��! #" : ukk = 0, uk+1,k+1 6= 0, · · · , unn 6= 0

'&/sT4wyx4T < �?D �

D�F 5 '%$�«�&�F 2 �.J�i� h��}O�02b[�J ]�_ E 's2&����� :

1–6

Page 7: A n B n B A A A~x - Ehime Universityrims-sym.math.sci.ehime-u.ac.jp/users/tsuchiya/math/... · 2005-01-13 · E 7DF yQ NTudb i $ % @ 1 G[ 2b) c T 0 HB # # .aL8 ! " $ %. i 3 P

���6 A 9$���¬#¼�½

p�-��4��TO� 3 �P$&% U Q�½P¶K/?C�'L)��sD E ��D#io&p

"�q�r A~x = ~b 2 �hz0� ~b ∈ Rn Q�R�/?T��#zOk ~x ��_�CA���������J� �

�.�Oi U ��R��P[��h2N)&chT 0 HB8�� E 'sH��B�sD

1.3 A �n�K� A−1 #��� ����E ��F�HB�Oi � � (2) ��h}O)BX�WOi A �4ZO$&% A−1 ��3P5+9m�>�� E 'b��y�sD

�O_u-����ya�Q�i46b7�$&% ')( ~ei, i = 1, · · · , n �i4¾ i [��2 1 HO�NX��E�� 0 'M8�B-���'L)u�sD�DJF 5 ��J�i U �bR��P[��2N)#chT 0 H�8 �9X�WOib"�q�r U~x = ~c �Oi� �· ~c ∈ R

n Q�_B�BT�k ��� E '?2&����vOTP���sD���Q ~c := ~ei Q�R�/?T.-�k�2�3P5)+������8���H#i���Xy ~si '*)+� : U~si = ~ei. � ~si ����$�% ' ( 8���H�i��OXy��+Qn ���#chC

S := [~s1, ~s2, · · · , ~sn]

�Oi n #!�"P$&%0H����MD E�E H�i US � )u�0'?iUS = U [~s1, ~s2, · · · , ~sn] = [U ~s1, U ~s2, · · · , U ~sn] = [~e1, ~e2, · · · , ~en] = I

'�8h� E '92&�����sD��9��Q�-�/ SU = I '�8�� E 's2B���NX�WOi�$�% S � U �NZ$#%�HB�B��'1�0x#�?D

o#p!P�02��yds�P��vNTO��� ��Q�'sv4T&��i SU = I �"!��u'�-

´�x&�42�i5�@�����+aUQb��#.Q!� �4)�� E 'O-�Hh:��sD±-0/ ��� �o�p

!��y�b�#�+e�$(/��ua�'L)+�b]Nt.Q�C�_h�B8 �MWOi5�3�����ua1Q!���4)&c.:N��-0/fX�8��ND����� H

3. (1) $&% U 2O� 3 �P$&%0H�����'�:4i US = I 'G8h�?$B% S 2.-�/ 3P5)#X�W�i S -9� 3 �O$&%�H���� E '�Gm�>#D(2) 2 �!O"�$#% U , S 2�� 3 �O$#%�H#i US = I 'M8#vNTO����'�:4i SU = I HB��� E'�GmJ>#D(3) U � n &� 3 ��!P"P$#%0HPi S � US = I ���CN)�' )+�?D E ��D�i SU = I 'M8� E '�4i���¡�g&%�'B¤�H�m�>&D#� " : ����i (2) �yJ�i n = 2 �-D��)(+*�2b[�J ]_OD� �QNi n − 1 ��H&(+*&2�[�J ]P_�'���}O)+�sD&)��0'Ui n ��bt�,��

U =

(U ~u~0t unn

), S =

(S ~s~0t snn

), U , S � n − 1 P$&% , ~u, ~s ∈ R

n−1

1–7

Page 8: A n B n B A A A~x - Ehime Universityrims-sym.math.sci.ehime-u.ac.jp/users/tsuchiya/math/... · 2005-01-13 · E 7DF yQ NTudb i $ % @ 1 G[ 2b) c T 0 HB # # .aL8 ! " $ %. i 3 P

'�(`d#'?iE�h}u�yJUS =

(U S U~s + snn~u~0t unnsnn

)=

(In−1

~0~0t 1

),

2&�#���sD�%�'B¤0���h}y�.J�i U−1 = S 2P��x#�4�#H�i E Xy �+vOT SU = I ������MD �

�B�0Q��.J�iP� 3 ��$&% U ��R-�P[���2N)&chT 0 HB8��9X�WOi U �4ZP$&% U−1 23P5�)+� E 's2�����vOC4D E#E H�i RA = U H���vPC E '?i R ��! �#$&%�Hh��� E ' ��A��)S'?i

I = U−1U = (U−1R)A, I = UU−1 = RAU−1 ⇔ R−1 = AU−1 ⇔ I = A(U−1R)

2#�P��J{i A ��!��0HPiN�N_ U−1R � A ��Z�$#%�H��B� E '?2#�P���?D��#�0N�y'&��0'siN ���}��02����.��X�C :

���7 $�% A Qb$+�N¬�¼h½

p���/MT&� 3 �B$�% U 2(� �1X�C`'1)`��Dª-

/?i U �4R��[@�02�)Bc�T 0 H�8P�h8 �GW�i U ' A �4! ��$B%�Hh��J�iB�&CA−1 = U−1R H����sD0C�Bª/Ui R ���O£���$���¬P¼B½

p �&)&!��&$&%0H����MD

1.4 �K�\� LU � $� F��PH#HPiG$.�b¬�¼&½

p'LZO$#%��b3O5n's������2&�P�Bv�CbD�$&% A G¬P¼&½

pQ���J^� 3 ��$#% U Q�½

p/UC+'�:�i U �b)#c�T#�bR-�+����2 0 H#8N�ª'�:PQ�`J A

�4ZP$&%0243O5n/?i A−1 = U−1R 'M8h�sD0C�Bª/Ui R ���O£��4¬�¼B½p �#)P$&%0H

�h�sD E X�� � A = R−1U 2&�&�0�42�i E �bF�H�� R �be�.h-ua �n/��K/Pd < T< �+a D-h/1i A � (1, 1)- [J� a11 2 0 H&8 �9X0WOib¬P¼&½

pP−mi1

1i , (i = 2, · · · , n), mi1 :=

ai1/a11 H ai1 �¦�¨hHh:��?D E �+'�:NiP−mi1

1i P−mj1

1j = P−mj1

1j P−mi1

1i , i, j = 2, · · · , n, i 6= j

1–8

Page 9: A n B n B A A A~x - Ehime Universityrims-sym.math.sci.ehime-u.ac.jp/users/tsuchiya/math/... · 2005-01-13 · E 7DF yQ NTudb i $ % @ 1 G[ 2b) c T 0 HB # # .aL8 ! " $ %. i 3 P

24[�JL]#_ E 'bQ���zn/4�+a D�¾ 1 %.�b[3�ub¦#¨#)������u��� �LX��� ���� � ������ ���� ��������A@��� ����� <������! #"%$�&('�)�*,+ �- ',./)��1032�4$5 ��687 '* �

M1 := P−m21

12 P−m31

13 · · ·P−mn1

1n =

1 0−m21 1

.... . .

−mn1 0 1

* � ' 9;:!<=">$?&�@BAC.>D � � a11 = 0 E3F�G � �,H 1 I 0,J%K �LI *NMPOQ�1'BRQS(+� ' 9 H 1 T + 0- U1VXWZY%��[ � �?� DB�\'�R(S�+ � � A].�^8_ [3�37 @ � Q1 := Q1j` � � Q1 := I

* �a� `�bQ1A

0dc�e �a�Pf � � $ [g� M1

0Bh>$ [ $ 5 '�*#i�j���@ � .(k�l ` *8& � �mI�T M1, Q1

0ml E @�n ��o�p �Q�>� �

M1Q1A =

a′11 a′

12 . . . a′1n

0 a′22 . . . a′

2n...

.... . .

...

0 a′n2 . . . a′

nn

*Nq�O �- '?)�*,+ 6 $'�.k�r ��sLt1�8u(v 0Bw yx ��� �

Mn−1Qn−1 · · ·M2Q2M1Q1A =

a′11 a′

12 . . . a′1n

a′22 . . . a′

2n

. . ....

0 a′nn

E @�n � A08q�O �- ',. �Qz � � Qi = Qij, i < j ≤ n,

` � � Qi = I,

Mi =

1. . .

1

−mi+1,i 1...

. . .

−mni 1

, M−1i =

1. . .

1

mi+1,i 1...

. . .

mni 1

1–9

Page 10: A n B n B A A A~x - Ehime Universityrims-sym.math.sci.ehime-u.ac.jp/users/tsuchiya/math/... · 2005-01-13 · E 7DF yQ NTudb i $ % @ 1 G[ 2b) c T 0 HB # # .aL8 ! " $ %. i 3 P

�� 'B.�� ` =� R = Mn−1Qn−1 · · ·M1Q1 �� ',./)�) �� Q2i = I � ��� ��'>* �

M3Q3M2Q2M1Q1 = M3Q3M2Q2M1(Q2Q2)Q1

= M3Q3M2(Q3Q3)Q2M1Q2(Q3Q3)Q2Q1

= M3(Q3M2Q3)(Q3Q2M1Q2Q3)Q3Q2Q1

E @�n � q�� �- ' E���Mi := Qn−1Qn−2 · · ·Qi−1MiQi−1 · · ·Qn−2Qn−1, i = 1, · · · , n − 2

Mn−1 := Mn−1

* f�� * �R = Mn−1Qn−1 · · ·M1Q1 = Mn−1Mn−2 · · · M2M1Qn−1 · · ·Q2Q1

* � ',.?kQl��@ A� � R = Mn−1Mn−2 · · ·M2M1Qn−1 · · ·Q2Q1

��@ l 3 �I�TU�8q�O��N��'?)�*�+ 6 $� � :

RA = Mn−1Mn−2 · · ·M2M1Qn−1 · · ·Q2Q1A = U.

)�)����Mi

0 o p ���� � c�e���'*��

Mi =

1. . .

1

−mi+1,i 1...

. . .

−mni 1

, M−1i =

1. . .

1

mi+1,i 1...

. . .

mni 1

*������ I(T1E���� � Mi

*gq 6/[ �"! )*L+ 6 $>'d./),��0�#$ � R−10�c?e?�

'*%�R−1 = Q1 · · ·Qn−1M

−11 M−1

2 · · ·M−1n−2M

−1n−1

1–10

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* ��'B./@ � �P := Q1 · · ·Qn−1,

(4) L := M−11 M−1

2 · · · M−1n−2M

−1n−1 =

1 0m21 1

.... . .

. . .

mn1 . . . mn,n−1 1

* o�p ��'*��

(5) A = R−1U = PLU

* � 5 '?)�*�+ 6 $>'�.\) E���� (5) E )�*80 � I�T A E 9 I�E M O�0�� nyA LU �1* !�n . I�T P � � I%E M�O 9�('�! � � O (permutation)

A0 � ����� IQT ��'�.�kQl>0 ` *3& � ��� E o�� 0���� :

���8 � � E n

����� I�T A � � A = PLU*

LU ��� � '?)�*,+ � - '�.��z � � U � l 3 �IQT � L � (4) E � E r 3 �IQT � P �dI>E M O10 � ����� IT � �'�. I?T A

+��!� �"�' ��& E R(S�# ��$!%�� � U E�& S('�+ �() �0� � !1)�*%�"('�.

*(+�, -4.) E/.>E c�e�0�0 1"2�3 � � I�T L

+(4) E ��� � �N��'�)�* ">$(&�@g.` �

L � ��� IQT ��?'�)�*30 � � � E ~c ∈ Rn� & �a� �!465 L~y = ~c

+�7 � � 098��)�*30 � �\)�*���@ �;:�< 7 @a. �

= p?> ��D ��� � �Q+ � A E LU ��� A = PLU+ � [ � � !%'>*N��'*%�@�!4�5

A~x = ~b+�A B� � 5 '�.�� ` ��� A~x = ~b

0 � �DC 6 ��� �

PL~y = ~b, U~x = ~y

0 � 5 �@ ! E � ('B. ) E 2� E ��4 5 � � I�T U , L

+FE �"GP�(l3 �IQT ��r 3

�I?T � E ��� AHB�%* 5 '�.

1 IKJMLON;PDQ�RTSVU!WYXKZ (partial pivoting) []\ ^ LUQD_!`bacL ^ed

1–11

Page 12: A n B n B A A A~x - Ehime Universityrims-sym.math.sci.ehime-u.ac.jp/users/tsuchiya/math/... · 2005-01-13 · E 7DF yQ NTudb i $ % @ 1 G[ 2b) c T 0 HB # # .aL8 ! " $ %. i 3 P

1.5 ����������)?) ` � E� � 0�� �$ � �!� �9��� � � I%E���� q���*�� �6�!4 5 A~x = ~b

0 �� )�* E�� � ����� ��� ��+�?'?)�*,+ 6 $ ��\*"! n .�)�) ` � � �$# I?T 5�%�* !n'&�( � # *)%) ��$� F�m./) EV. � � �,+ ` � E�-� 0-. ` j�) I?T>E�I�T 5 � �!/)Pi�j�)�0�@�n .�1/2 E�3�4 � � � � I(T 5 ��5 r � 5 � o�p � �6)�!�) � �-7HEE @�n � o�p �() - � E $?+���+/8 * � � $���� � � � *) !%'�. ` ��� I(T 5 E�op E E D E +�� � O (permutation)

0 #$ F� � �%� � !1D E �H('()�* ���:9�; ��<0/8�����+ ` � ` ��=�j�)>� ` n@?BA ���'�./)Q) � � 1C2 E�3�4 � * ��D � � I?T5+FE(�6) -�GCH 0�i1j � ^ E @�n � I?T 510 o p �O� [ E EFG/H 0�E � �>$�0,0C)!1)�n .k�r I?T�E�I�T 5103i�j�) ! � ��& � � n

�6�!� I�T A0

n E,I?U�V WZY ~a1, · · · ,

~an ∈ Rn0 J � �K) 9ML U1V W Y �N�K) A � I �FO�) �>D E *#i�j�� IQT 5 � n PE8U

VXW Y ~a1, · · · , ~an

03q*Q$�8��'SR�Q�T��FQ�*Ni1j?@�n:

A =

~at1...

~atn

= (~a1,~a2, · · · ,~an)

t ,det A = det(~a1, · · · ,~an),

det : Rn × · · · × R

n → R.

)�)���UH��_P+ I�T 5 � “ V�W � ' ”)�* � ��� E 2

� �"(':�

���

(a) A ��D�I�T A−10�8 �

⇐⇒ det A 6= 0

(b) det A = det(~a1, · · · ,~an) � � U�VXW Y ~a1, · · · ,~an

0�X!*#��'Y I n Z:[E “ \�G � - ” [^] 0 � �m.

G�H (a)0 � ��)�*a+ � ) E = p�_ 3 W Ea`cb�z /�3. G�H (b) � � I�T 5 E “ d�e�f�g ”

o p �%D��',. B,h U�V W Y ~ei

� & �i)�� ~ai = A~ei

* ��' E � � I�T A��@$� o p�kj�'

Rn$ [ R

n l E � �nm^o�� f !�)�� ~e1, · · · , ~en

+�p>'/B,hn� [� ~a1, · · · ,~an0�X�*#��' Y I n Z*[ ��q �kj�' 9 k�r E�r 0 s/t�A .�) E Y I n Z:[ + “ u jv) ”

!j>� �

A��@ � o p �kj�'K� �Cm*o � 1 & 1(one-to-one)

m^o� � � !1)�*���� ���(Kw D m*o �,x�y ��� ! . ` ����) E * -FD�I(T A−1D x�y � � ! . D � �1) E Y I n

Z*[ + “ u jn) ”!�� 5 j�� � D m�o!* D IQT D x*y ��'�)�* �"��'g. � ` ��� G�H (b)

E �/z +^{-| g � 6 $-j� � G�H (a)+F} ��~"��)?* � < < �"��' E ��?'B. � f � Y

1–12

Page 13: A n B n B A A A~x - Ehime Universityrims-sym.math.sci.ehime-u.ac.jp/users/tsuchiya/math/... · 2005-01-13 · E 7DF yQ NTudb i $ % @ 1 G[ 2b) c T 0 HB # # .aL8 ! " $ %. i 3 P

I n Z*[ 0 “ u j�' ”)�* � ����� g �/� n

� [ �3q"�� - '�* - ��E E Y I n Z*[�E[^]$� � �!E n ��� !\* -"�,[*] ��� ��('>*#��'�.

PSfrag replacements

A

`8b ��) E�� �(+ } � ~ �1)(*30dS� � � F�G � I�T 5 det+SE��8�!) - G�H �8��

6 $�'Q)?*30a"� ��@1nZ. BFh I�T I� & � ) � � I = (~e1, · · · , ~en)t

�6 � �~e1, · · · , ~en+:p�' Y I n

� [�E�[^]$� � o p @�� 1�"?'�./@ *) �

�� ��(I) det I = 1

+-} ��~���)�*�+ R(S �)�*�+ 6 $'�.� � ��Dn�~a1, · · · , ~an ∈ R

n E > ��� � !�D E +� @��*��m@1nZ.�) E� � � �!�F45A~x = ~b ����� Q E Q n

��5 E Q n− 1 E � ~F7��F��465� E ��� � + x*y �%� !�@n ��63�� ~b+ x:y ��',.�l EFGCH (a)

0 I�T 5 det A+FE �P�!*#��'*%��) E� ��

det A = 0* � *)�! � 5 j�� � [ � ! � b � E ����� ��(II) det(~a1, · · · ,~ai, · · · ,~ai, · · · ,~an) = 0

+-} ��~ *)�!%',R(S(+ ('?)�*�+ 6 $�'�.� � � U1V�W Y A = (~a1, · · · ,~ai, · · · ,~an)t EmH i I ~ai

0 o Q α � �K��� � ~ai

� U1V WY ~b

0��/j?'?)*808i�j?'�.2��� EmF�G%E�r 0 � !C):01'* � {~a1, · · · , α~ai, · · · ,~an}+:p>' Y I n Z*[�E�[^]�� � {~a1, · · · ,~ai, · · · ,~an} E p�' Y I n Z:[�E,[^]�E α � ���� E�n z � � {~a1, · · · ,~ai +~b, · · · ,~an}

+*p>' Y I n Z*[�E,[�]�� � {~a1, · · · ,~ai, · · · ,~an}*{~a1, · · · ,~b, · · · ,~an} E p>' Y I n Z*[�E,[�]�E�� �"� � E�n z * !�n,)�*�+ 6 $�'B.� ` � �F� E����� "!$#�

��

(III) (i) det(~a1, · · · , α~ak, · · · ,~an) = α det(~a1, · · · ,~ak, · · · ,~an)

(ii) det(~a1, · · · ,~ak +~b, · · · ,~an) = det(~a1, · · · ,~ak, · · · ,~an) + det(~a1, · · · ,~b, · · · ,~an)

1–13

Page 14: A n B n B A A A~x - Ehime Universityrims-sym.math.sci.ehime-u.ac.jp/users/tsuchiya/math/... · 2005-01-13 · E 7DF yQ NTudb i $ % @ 1 G[ 2b) c T 0 HB # # .aL8 ! " $ %. i 3 P

+-} ��~ *)�!%',R(S(+ ('?)�*�+ 6 $�'�.*(+�, -

5. 2� � E8F�G l E ��� � � G +F} ��~�������\n,)�*80 � r 0 � !�)8">$&^):01@a.

kQl � # )�n !�n G�H +F} ��~ ,)�!�)���� !��-%%* ! n=S�� (a), (b)$ [ � # (a), (b)+-} � ~ ^)�!�j��"�>)�n �� :)�!%' � b z @��%/* !�n GCH (I), (II), (III)

0“ !)

”06��.�� ` � �

(I), (II), (III) � � (a), (b)+F} ��~"���(& E “

R?S $F% ”�"�� �)"!

'�)(*B+ 6 $�' 9�� � � :"<a�K� 6 5 � � � !�+ A .^R � �?) E (I), (II), (III)0 � IQT5 E o�p � /&��i) ! !�)�*�+ 6 $ *)�!�',.�� ` � � I?T 5 E “ ��H ” � � (I), (II),

(III)�� ���� E GCH���� )/) (I), (II), (III)

$ [�%-���� )�*B+�� - ' E ��?'B.�kr �1)Kj10��C)*0�@�n .` b � IQT 5�0�k(r E @1n � - _�� * oPp � @�n :

���9m^o

det : (Rn)n → R+�k(r E 3

� EFG/H 0 E � � * - � det0������

(determinant)*��� .

(I) det(~e1, ~e2, · · · , ~en) = 1

(II) det(~a1, · · · ,~ai, · · · ,~ai, · · · ,~an) = 0

(III) det(~a1, · · · , α~ak, · · · ,~an) = α det(~a1, · · · ,~ak, · · · ,~an), ∀α ∈ R

det(~a1, · · · ,~ak +~b, · · · ,~an) = det(~a1, · · · ,~ak, · · · ,~an) + det(~a1, · · · ,~b, · · · ,~an)

! �(D .*j"��+���l E�$F% (III)0Fm�o

det E ��� �!�# * ! nZ."� � � �$# 0&%('� ' ~ F ��� �� [ � �Pl E (I), (II), (III)

0iE�� � @1n �^m�o�+�7 � � x:y�� '?)�*0 ��� R�S�+"('m+ �%)�)�� ��$!% 0 E � � m*o+ x:y)� '>*+* o � )",�<�0 � 5 )!1)�n .` b � $F% (III)

0&* o�� '�*%� $�% (II)* $�%�� ��(II’) det(~a1, · · · ,~ai, · · · ,~aj, · · · ,~an) = − det(~a1, · · · ,~aj, · · · ,~ai, · · · ,~an)

� � s T ��('?)�*�+ �.-d6 $'�. ) E $F% (II’)0 � I?T 5 E�/$0�# *�!�nZ. R21��

1–14

Page 15: A n B n B A A A~x - Ehime Universityrims-sym.math.sci.ehime-u.ac.jp/users/tsuchiya/math/... · 2005-01-13 · E 7DF yQ NTudb i $ % @ 1 G[ 2b) c T 0 HB # # .aL8 ! " $ %. i 3 P

(II) E ~ai

�~ai + ~aj

0 C�� � '>* � (III)0%# :)

0 =det(~a1, · · · ,~ai + ~aj, · · · ,~ai + ~aj, · · · ,~an)

=det(~a1, · · · ,~ai, · · · ,~ai, · · · ,~an) + det(~a1, · · · ,~ai, · · · ,~aj, · · · ,~an)

+ det(~a1, · · · ,~aj, · · · ,~ai, · · · ,~an) + det(~a1, · · · ,~aj, · · · ,~aj, · · · ,~an)

=det(~a1, · · · ,~aj, · · · ,~ai, · · · ,~an) + det(~a1, · · · ,~ai, · · · ,~aj, · · · ,~an)

+ 6 $�' E ��� (II’)0�� ',. D � � (II’) E ~aj

�~ai

0 C�� � '*%� (II)0 � '�.(kQl

��� I�T 5 E�oPp � � #(I), (II), (III)

%\* #(I), (II’), (III)

% E ^3_ [ 0%#� ^)%D�@�!)�*,+ 6 $� F�m.) E (I), (II), (III)0

P λij, Qij

�dJ�� � '*��

det P λij =det(~e1, · · · , ~ei, · · · , ~ej + λ~ei, · · · , ~en)

=det(~e1, · · · , ~ei, · · · , ~ej, · · · , ~en) + λ det(~e1, · · · , ~ei, · · · , ~ei, · · · , ~en) = 1

det Qij =det(~e1, · · · , ~ej, · · · , ~ei, · · · , ~en) = − det(~e1, · · · , ~ei, · · · , ~ej, · · · , ~en) = −1

0��\'B.�� ` �����

��(c) det P λ

ij = 1, det Qij = −1

+ 6 $ ���. ` � � A+�r

3 �IQTz *�� � i = 1$ [�� ��% ' � )"! � *

det A =det

a11 0...

. . .

an1 . . . ann

= det(a11~e1, a21~e1 + a22~e2, · · · , an1~e1 + · · ·+ ann~en)

=a11a21 det(~e1, ~e1, a31 ~e1 + · · · , · · · ) + a11a22 det(~e1, ~e2, a31 ~e1 + · · · , · · · )

=a11a22 det(~e1, ~e2, a31 ~e1 + a32~e2 + a33~e3, · · · )

* ��'B.A+���l

3 �IQTE8F�G D s,t ��?'B.�)Kj�0 � 5 )"! � * �/� E G�H (d)0

��� )�*�+ � - ',.*6+", -

6. A+�l

3 �I�T � ` � � r 3 �I�T>E$ ��� E 5�+-} ��~"� )�*80 ��7 .

1–15

Page 16: A n B n B A A A~x - Ehime Universityrims-sym.math.sci.ehime-u.ac.jp/users/tsuchiya/math/... · 2005-01-13 · E 7DF yQ NTudb i $ % @ 1 G[ 2b) c T 0 HB # # .aL8 ! " $ %. i 3 P

�(d) det

a11 . . . a1n

. . ....

0 ann

= det

a11 0...

. . .

an1 . . . ann

= a11 · · ·ann

l3 �I?T U

� & �i) � ����4�5 U~x = ~b+ � � E ��3�� ~b

� & �i)�7 � � 0�8 �R(S�# ��$�%�� � U E & } � +m2 ) 0� � !�)�*%� � F� ./@ :)��

det U = u11 · · ·unn 6= 0 ⇐⇒ U E@&� } � +m26) 0��� !

⇐⇒�!4�5

U~x = ~b+ � � E �63 � ~b

� & �i)�7 � � 0�8��+F} ��~"��)(*B+ 6 $������ � � * D,l 3 �IQT �*� �K) � G�H (a)

+F} ��~ ��)�*B+6 $ F��.�S)@� � �

det(AB) = (det A)(det B)0 � �i)*0�@�n .:L U1VXW Y

~ati := (ai1, · · · , ain), ~bt

i := (bi1, · · · , bin), i = 1, · · · , n0S& !�)��

A := [~a1, · · · ,~an]t, B := [~b1, · · · ,~bn]t

* oPp)� '*��

A :=

~at1...

~atn

, B :=

~bt1...

~btn

, AB =

a11~bt

1 + a12~bt

2 + · · ·+ a1n~bt

n

a21~bt

1 + a22~bt

2 + · · ·+ a2n~bt

n...

an1~bt

1 + an2~bt

2 + · · · + ann~bt

n

* ��' 9 : < "%$�&�@BA ./@ �)�� I�T 5 E � � � � G 0 # nm*��det(AB) =det(a11

~b1 + a12~b2 + · · ·+ a1n

~bn, · · · , an1~b1 + an2

~b2 + · · ·+ ann~bn)

=∑

det(a1i1~bi1 , a2i2

~bi2 , · · · , anin~bin) =

∑a1i1 · · ·anin det(~bi1 , · · · ,~bin)

0/�\',.�)�) � � � ∑ � � ik, (k = 1, · · · , n) E 1$ [ n E 26) E T � 6 � *)\*'�D E * � ',. I?T 5 E-GnH (II)

0-!�! ��� * ��Dv� {i1, · · · , in} E ��$ � s�� D E+�-j �det(~bi1 , · · · ,~bin) = 0

*��%' E � � i1, i2, · · · , in+ 2�)����>' F(G ��� ` �

{i1, · · · , in} = {1, · · · , n}* ��' FQGz 5 i�j�j��@�!1)�*B+ 6 $>'�./@ *)��

det(AB) =∑

{i1,··· ,in}={1,··· ,n}

a1i1 · · ·anin det(~bi1 , · · · ,~bin)

1–16

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+ 6 $� F�m./)�)���� {i1, · · · , in} = {1, · · · , n}� E ��� i1, · · · , in � 1, · · · , n

0FO!)� j��>D E � (',./) E @�n ��O!) � j

(6) {1, · · · , n} =⇒ {i1, · · · , in}

0 �n� E���� (permutation)

*��)� z )�*80F!�! � E nZ.�l E@� O�0(

1 2 · · · n

i1 i2 · · · in

)(1 → i1, 2 → i2, · · · ,

* ��- O j�'B. )* � - � ` � n

� E@� O�2 [ 0 Sn

* ���E + C Q f � E���� ��� F�m.�S)���l E@� O (6)08i1j�'* - � I%E M�O Qij

0%# �)

(7) (~bi1 , · · · ,~bin) =⇒ (~b1, · · · ,~bn)

E @1n �FO�) � jQ'�)�*B+�� - ' � b ��?'B.�) E * - � det Qij = −1� �"� � '�* �O!) � j

(7) E 1 � � Qij

0:Q���# n $��� *Q���# n#$Q+ � S � ?'()�*B+ 6 $'B.� O σ ∈ Sn

0FO�) � jQ'"1 � I%E M O+�*Q���R(S�� [ ��� σ0�� ��� *���� � I

E M�O�+� *Q��mR�S�� [ ��� ��� *��� 2. E �i)�� � O σ E���� sgn(σ)

0

sgn(σ) := (−1) ������� (7) ���! #"%$'&(�*)�+,��-'. =

{1, σ

+� � O−1, σ

+� � O* o�p � j��"�

det(~bi1 , · · · ,~bin) = sgn(σ) det(~b1, · · · ,~bn) = sgn(σ) detB

+ 6 $�'�.(k(r � � O10

σ =

(1 2 . . . n

σ(1) σ(2) . . . σ(n)

)∈ Sn

2 /1032 N545687σ2�9�:(;1<>=@?BADCFE=5G87=IHKJL2NMPO�QbLSRO` [�TVUBWKX C I JYC14RBR[ZN[\^] U J dV_D` = IKJ[a N845b>cD;�d�e8fhgi28;YjLk3:,;3l�monTL d

1–17

Page 18: A n B n B A A A~x - Ehime Universityrims-sym.math.sci.ehime-u.ac.jp/users/tsuchiya/math/... · 2005-01-13 · E 7DF yQ NTudb i $ % @ 1 G[ 2b) c T 0 HB # # .aL8 ! " $ %. i 3 P

����� � ��������� � ���� �

(8)

det(AB) =∑

σ∈Sn

a1σ(1) · · ·anσ(n)sgn(σ) detB

=det B∑

σ∈Sn

sgn(σ)a1σ(1) · · ·anσ(n)

���� � ����� B = I ����� � det I = 1� �

det A =∑

σ∈Sn

sgn(σ)a1σ(1) · · ·anσ(n)

���� ����� j���� (8)� C�� �i)��

det(AB) = (det A)(det B)��� ������! )"��

�(e) det A =

σ∈Sn

sgn(σ)a1σ(1) · · ·anσ(n),

(f) det(AB) = (det A)(det B)

"�#�%$ � � � � � 3*4 �'&�( � � � (e)� �*),+"-

A�.+�-�/ �02143 j�!5 � � �

6'7 ((e)��8:9;�=<�>@? �A�B� �=C 1�D.EGF &�( ��H�� 6'7 (I), (II), (III)

��8(e)�

9?�I ��J#K�L2M �=N � & �!O'P RQ �=S�T�UWV 2X�Y[Z�\ ��� 67 (I), (II), (III) (e)("].^B&_ ���"� I � 6'7 (e)

( � 6'7(I), (II), (III)

�a` I2Y�bdcdet : (Rn)n → Re�f"g"h%i�jWg�k Y�l�m�n!o J.p g Y�laq

r's�tdu7.+'-

Ao!vxw�>"y +�-;/

det Aj

(e)& 0@1zwaI�n{Y[l�n}|@y

det A( 6

7(I), (II), (III)

j ` I2Y[m�n j X�~�qm#m�� &�� l'n�y +%-#/e 5 e��Be 6%7 (a�'� o�X@Y�m�n�� & |#l�qa�!�"y,�;> e�+#-

A( y

A = PLUn

LU �d��� |dI�m;n j!� ?� 5 \�q��%I 6'7 (c), (d)Z*�

det L = 1,

det P = ±1, det U = u11 · · ·unn � _ l'm�n!o p g Y�l�nay (f)Z��

det A 6= 0 ⇐⇒ Ue v��%� � �2��> 0 �'� ?

⇐⇒ ��� / A~x = ~b�!� g�ed� E�� ~b

o!v�w�> f%g � jW�������Bl e � yd�;�¡ £¢¤n{Y�l 67 (a)

�dX43�<BI2q

1–18

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��I���� /�e 6'7(c)Z*� y

3�����

L, Uo!v�w�>��

det Lt = det L, det U t = det U�}� ��� � m;n���8A� � _ laq�� e ����j� Y���� PoWv�wA>��@y

Qtij = Qij

n 6�7(f)j������ y

det P = det P t���"��l�q�Z��#>%y

det At = det(U tLtP t) = det U t det Lt det P t

= det U det L det P = det(PLU) = det A

�����Bl e � y Q e 6'7 �2X 3�<BI :�� ��(g) det At = det A

��� /�e���� 6 n������! 6(II), (III)

j"� \dn�y��!� /Be�#�$ o&%�' �)(+*�, j��� o.X�Y[m�n�� � |�l�q��2��y

det

a11 a12 . . . a1n

0 a22 . . . a2n

......

...

0 an2 . . . ann

= det

a11 0 . . . 0

0 a22 . . . a2n

......

...

0 an2 . . . ann

= a11 det

a22 . . . a2n

......

an2 . . . ann

o p g wdZ�\�q�m�<-�%y%? L ? L%�/.'�*��0 �21�l'mBn.� � |l 35476 0 �81%> �"93�?�: q 38.o�y(II)’

j�� �">%y

det

0 a12 . . . a1n

a21 a22 . . . a2n

......

...

0 an2 . . . ann

= − det

a21 a22 . . . a2n

0 a12 . . . a1n

......

...

0 an2 . . . ann

= −a21 det

a12 . . . a1n

a32 . . . a3n

......

an2 . . . ann

n�?¡\�m'n J �#��l�q;�< j �� o.Y�l}I!1#o2y=��� A��8?>

i�!>

j� j)@ ?�I

n−1

Q�A ��� j Aij � � Y�m�n}o!Y�l�nay�B e m�n"�

det

a11 a12 . . . a1n

0... A11

0

= a11 det A11

�DC n=E � m�n�� � |�l�q�F�G ]"H o"IJ1@>"? � n

1–19

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'

&

$

%(h)

a11 a12 . . . a1n

a21 a22 . . . a2n

......

...

an1 an2 . . . ann

=

n∑

i=1

(−1)1+iai1 det Ai1

�d� � � � mBn��'���*l�q�m'm � �"y�> 1� ��(-* w�I'�"y f � o)> j

� � (+* w�I� y�"I(g)j�� \dn=>

i� � ( * w�I4� Y�l'm�n J}� |'l'm�n����"�Bl�q

r;s�t�u8.B n ]/H o�w�>�y �!� , j > j

� � y _ l2?8��> i� ��(-* Y�l�� , j9���q

1–20