Matrices
Matrices
Matrices
Oi
A
Q
O
A
ay
ymmeiL Mal
Said fo be mmetic
mahiG ss
A Squoe
amahi
s saiol fo be Symme
fh the otherwovo the obqo ol
all he elemens abone nd below
coith there posPAorn
eSame
3
S
A 2 3
A13
Kau mmerc Moi
Squa motiy ssafo to be Shew Smmei
-Oji
A
5
O 2
3 3 3
a2-S
2Nlpetent MatA
ASqane mah' A_called nflpotent mak? theve
xSt a osffe nteges 'n Sych hat
wekutay Mahcs
maix
Assalo to be fnuslutau
oIndennpotent Mai S
maic A' fs indopotopt Makit i
uar called
A A
Hexmi Eian Modis
fn ohsch (i, D}"
Dlement Ts qal the Confugobe cmpler
elenent aj = 0J;
he (
air
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A (314i) 6
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O21
o p n a _ are_pmely fmmoharq 3
dtous heml
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Z A mxn then hebspoye
6 A is clcno tec b A ox ohich s
amabic c Oxoley Dxm
A a
O3
12
O3 O23 a 33
33
h o qonal Mot ies
A oakirs salleel_oytheqanal makis i
A-AT
e ddilRe Suhshe tf on fuo_/ofs
r t o o mabees Same xoley
hen O.ddifion c fhese mabire
A O12 b3
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O3 a32 b31 b bs
b bi bis
b b3
a 3 a32 a33 b3 b3
b
AB
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Mul f6/fcaidn C
Lte assocfa Hue
Afs mafs then
AT) - A
T
A o ue makrey some_otie
hen
ATB
A BT=CA
MineY a atv%
A is o SquaU mairs and ait fs an
Plement Lchfch s n the YC_ancl J *
Cekiumn hen deteminant efthe Makix aft
deletin ISallecl minor
CThe elemCnt
A a13
Cs
3 azz z3
M
A32 Qs3
Ci2
a3 Cz3
a 33
a 3
ino
a22 23
3
a 3
Dekxofnant 0 _ aMabix
a
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Oi
31 32 O33
a23 a3 t O i 2
2
a3 ai 32
3
O3 21 Q3-a22 3i
d Rit the ore
e O He
n vevse mobf
mo ohich fs hon-
then fnunse
A cdi A prsufdeol 1AfD
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a13
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as a33 31 03
a2 O3 as
q22 A23 azr a 23
dfoin o a mois
Tanspose uo-acoy sCalled Adjetnd uA
oke
T3 Ts the fhuese then AT
x_
Ocle
A nfq
uesse a o moth
hen
Let Poses t u e
-O
A=I
Comb HOL O
AB AC
A B
(A B.
GAhd h e 7nuens o s the m a i s
A
2
A 2 = 4 -3
2 -
Co-fotor A
- -
asjon fA_ 2
Qcla 2 1,
A1
LA _GS/1-4)-2/2- 4)
+2
(4+2)
3t + 12 = 27
6
-4)
-(-2-4)_ C-4)
6 +6 -3
6 3
3
adj A 3
27
an of q Malvirs
fub matu
ebi,
A 9s fhe fype MKD dhen a
mafy Swy Cslouog 7 bsth
bioned D eauing Sum vou
caq
-
Qy
3 O343x4 eLins,A
C
22 a3
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C u Qu an
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abaties hohiy
Kank of a Mat
(perite Pntemo
a mah A
posses olCoufr q
biepeies
Sus
Sauoren mahiz
AA dey G)
(4
Sfrgulcn
FA).
in 4 enenal fe)Emin(,
nKanR
n e a null mati fs roys Jev
s armathix e Ovder nxh
A
Minimuwm yan R
SO) =
A) =
ovder
0hon-Sin9lo moT
i d
DXD Hen
SA)=n
amakin A then
A s the assose
SA)=Je)_
mot i
Konb moi
be
A
SA) nin(, 3)
A) = 1
he vanR
A 23
O 2
AL(6-3) - ( ) 3 ( )
Hene Pon 3
fhed e
3
Leeniy cu Opeafian Metheo
Let (onsTdY o ma
A 92 C23
Osi a32 O53 s
A 2 2
25x3
P P 2
3
C
2
R 6
Her
fhe ane /hree
SA)=
A
2
R 2
Orehy ihscl
henspOs? chonge de Oy Cokeun
A
2
R2R
3
R2
- 2
ten. c nvecs hy ee
noxma Zoxty
Pncory sih
2
)Don STqula Mohi
he ol,i
ned he nonS Ca mouhe feY
cOn be w en
A
ALc
o
o o
R2-R o
R3-3R) -2-2
2-2
A3-
b -2
A
2 -1 1
C2-C
O 2-2 O
C3-C -2-1 J
- O
-1
-2-
-1 o o
O: O
- O
AVexse of a maticS b
by
E Ro ansfo mafien Ethoo
Lot A
Ps a ahe tfovde D
hfh fs DonSfotLa hen
A LA
I Shews hat A = B
A 2
aoli
3 2
2
Can be oi#e
A=1A
3
23 A
2 o o