Matrices are an ordered rectangular array of elements called entries. Common types include zero matrices, identity matrices, row matrices, column matrices, square matrices, diagonal matrices, scalar matrices, and unit matrices. Basic matrix operations include addition, subtraction, multiplication, and finding the transpose, inverse, and determinant of a matrix. Trigonometry defines relationships between angles and ratios of sides of right triangles using functions like sine, cosine, tangent, cotangent, secant, and cosecant. Standard trigonometric identities and angle sum and difference formulas are used to manipulate trigonometric expressions.

1. Matoices
An rdezed ectangula asray of elemens is calteca
Matais. Ne eonLine to maeees wñore elemens alL Veal
c o n p l e numbers.
pes Mateiees
ONall matricesC)Zeso Mabiz.icach element zero
densedy Loo
oO JBx3.
E C 2 3) ix)
Ro matrz' one o e
3Jax
column
colwn mataiz: l one
g a i e mafai No.of rous &columns a s e egual
2 3
Teace da nafziz e Su7
f hediagonalekmens
+H
a sguane r a t a i r . /
ulas matrei: No Raus &columrs ase no egual.
. Bx 2
Digonal mafu:sgpuar atrir aheh the elemens
encept he doagonal elemen are peio.
Scalur mataz ln ths diagenal elemenk air cgual
oo7

2. unit mate /deniynai Yn gaiu masna all h
ekmens
q diagonal d egecal 1 & itcsdenstd by
L /001J
o) Tiangular maix A sguass
matrix hose
elemens a i eiha
ahe /heloro ha leadiy iagonal
a l e call pelo
es
known
a a
m a t z .
takangaler 3
2O
IA3
ansposs raiz nterehasg
*hdditonEsuboactn Amasee the two ma ties having
d e diensions
A= 4a 7, by ba7 Ath t h aha
coperties Addiken 9 matYiees
cemma fative prepea
AB&o ciatve propery:
A+ CBte) - G t 6 ) + c .
A+B BtA
AelcdttveLdnfi t o =otA=#
Addhve en erse
A+S =B+A =o.
Multiplicatim Cprodaet) 0atmces mullype Totox coen,
K7.

3. >opeyti dmulpbcafin ares
Assocate laus
:AB. A c
Distribuive lew: ACBte) =A:8 +A
(A+B)C Ac 4 Be.
Cemulafive law: AB= BA.
ransyause mariz
237
hen
AGmxn
Ahen ransprse
matnz i At
perties honspos rat
AT=A
RAY = KA
a+e)= AT+eT
Ae) BTAT
ymmerix matir A &guaw
matir A said o he gnamemie iL N=A.
Skeus kmebi
6kcw
y m e r i t m a i z : A
sguase mair A saod to be
rerir AT= -A
*Detaminant : Ltas denotrd by A/.
Tx cdeemmants A5 A/=S
nts b =IAl = acl-.
2x2
Mino: Me aathirminor nekmnt a a marix adekerniant
tha sh mam cmaton by emevin the ros hcalomn
Cn
A

4. mnaret
element = aa |
a E ) t 1
C I ) ' * a
Cofact o an elemen aij of a
matrir s the mines
Cefactie
maloiplice ( - ' 7
Og
Deesmeran da mair d ne let A [a] be the matrir
a e r d e r ,
then dehrminan
depined to be equol to a'.
zoperhcs ofDeferminank
Of eah etmen a r o w columnSguasa matra
gero
then he de erminans d he
m a i o Ggea)
A444=le 61,. IAl =
e
t w o
raws colamn t a sguat matmr au idental, hen he
determinant of ma trix d Og)
b c
/Al-0.
A
twv 7eous d
cotnn n he marix
d r
intrehanged
then
h detexominant changes
cacdcacheement a hod column a sgcuarematrix is mult
phed bya non zrO numba k, Then he de ksnminant the ma1 r i
oblrined
k imes h defeminant ofgin mahi

5. ka, kb kg
a lel k A.
Lag
b
A
k
corespen ding eemens o trwo sorurs eo loumns qa
squasp
mair a t he 6ame atb, hn he detrminan
ha ma iz e
L each element aa mos de colan 2a
Sum two numhess, hen ik detesminan can k enprcsed
es he 6ume dettsminank to5ua Mati
by t t
cla h
A
b
b
X coach elenn asmo acolunn a Z
Coeponding
elenehs anour r Crcolun
wth k mes he
egual valer he d terminant f the
matix
C
hen hu Kalues ef ho dekr muinant o esuly
m a t n
Ta,tka, btk
b C then el=1B
a by
C
A B
b
Ehe sum e he poduct f the elemenbs s ay s icolumn
asguara
m a r e t h he cofaef he corespondir
elemends
anohey o o
decaemn The mahr es zeio
) Per any e a s
matri A, IA| = [A"7.
a b C
A b2 2 hen A"= b, b, b
b
b
Le s
b

6. Note (i )lAB = IAllel.
(i) A Amariz aperarlaur) then h deker minntqAl
h produefd diogonal mathiz
&Adoint inver ae a matru.
->Singalay
and Non-singudarMatir: 1 square matri aiolt5
bendar if ik de terminant e zeto othercoise t u saidts ba.
o SNgulay enaix
let A be a square
matniz. he mamx B b abtaned
Cefaede af
epla cay
tha kenb q A i cofacders
hen he mabr
B es Catled
matiz aA
o n t ama
let A be a squate mati han h rarapse fcofacte
marir A called hyoint qA.AA
denotd)
Lnverse a matr
let A be a éguaze mapiz. al y
that A berfble
a m a r
e n s k Sueh Kat AB = BA= V ehere Pesanit
ma Aiss n o r -Sgnelay mabi e A Ezb.)
A= adjn)
A
TAlF
Cammers Kul
Conaicer he Cncar
aytby t3

7. Ax B
he m a i representahon
A Coefhèient matria
d
d,
ds
a b 4
B Constant
cSolupion
A a b
A
.
1=/lo b
yb
ba
a, b d
du
b
Az
da bs
lnversion rethod
Matrir
me hod atia
mahir foam
Ax- BO
A a, ba

8. Irigonomety
eignome reic hatios
Coeradnete Sysfem:
90
Sin ie te All retve{
Cosec 360+O
90-6
80-
r e
80+o 240+0
1anet
240-0 360-e
aret
See
Cot
270
*We apgk e è +Ve ip itu measurd in he erticlockusise
clirecton &ve it s
m e a s r e d
n the glockesce dirceticn
360 2.
-90-4 80=T270
edgomerie FunctionsCRatos) Angles
opposite AE
A
potheses
6in
oppas dypotheses.
side
cos acent Be
potheses
AC
an s i k
Acjacent ypohere
Seco-
Adjacen
Sicde.
helacenF
Ceseed = y theses Ac
OPposif AGcotx #jaent .
AB.
Psi

9. Standasd Reults
Sine+ Cos*e =/
Sino I-cose
'+tane-Gee9 lo) secto- tans =
+ Coto soseee ) Cose e29 -
Coo:l
Sine CeseeS =|
c a s o .
Sece =I
tane. coto=l
Sine tane
Cose
cosg cote
Sino
Sine =
Cosec e
C o s
Sece
. tano Coto
Secs+ tane
Secetane
CosecS tcore
Cersecs -cote

10. Valaes ef the teigonome trie fune fions f he given angles.
26° 276|2T 20
Sine
Cos 9
|tan o O
2
2
see V2 2
Cotco 0
convezsicn f Degree & Radians
Radians Degpeee x
Degeee : Kaians x d
TT
Sinco) - 5 i n G
Sec C-®) = +Sece
Ces Go) +cos
Cotl-6 = -cof
tan ce) =
-
tano
*Tofind he ange 4teiyoncnehie foa:
Weik he angk f teigonomie foms in the foan o
e ) ter n'is e ten there à ne ehapg e in valu
n ' s odd (Stn<>os; fan A cot; cosee
t ee)
e
ighVeftve) of he angk depends on he cange in uAich
&uavdioF les

11. agorou heorem
AB+B= Ac
Cempeund fngles
he algeboeaic sum o
two e me anges as Called
Conpeund dngles
Sin A+B) StntcoB +cosASinB
2.SinCA- B) : SinA cos8-
costsinB.
BCas CA+B) =
CasAcosB
-
StnAsinB.
4 Cos CA-B)
=
casAcosB
+ S i n A s i n B .
tan CA+e) : -tanA tanB
I-tan tan8.
tan t tanB
1+ tanA tanB.
6.tan CA-6) =
7. cot CA+B)
= Cot ACot-1
CotA +otB
8. CotCA-B)
CotA cotB+/
Cot A -
CotB.
9 6iA+B)+SinCA -A = 2$inA cesB
|o SinCA+O) -
sin CA-B) : 2cost SnB
Cos A+B)tcosCA-8) 2casA CesB
2 CosCA+B)- Cos Ct-B)
= -
2sinASinB.

12. Cos(A -B) -Cos CA tB): 2sinA Sin 8
13- SinCA+6) SinCA-e) =
SinA -Sin2Bz cosBacosA.
|4 CosCA+B)
cesCA-B)
= c o s - s i n A = cosA -Sin2e
Tan+n)
= Tan cosh tsinA
/-tanA
/ttanA
CesA tsinA
CosA - SinA
an - tonh
CosA-sinA
6. tan C-A)= + tant
CosA fSint
S i n t SinD = 25in( cos)
18. Sinc-SinD =
2co si )
9. CesCtCosO:
s c a ) c o r )
20.cos
-
c o s D s -25inCP) sin( )
Maltiple and
ub-multipe tles
A é angle fhen 2A, 3A, YA... .
ete. ere coled ultipls
A a n ang hen are Sub-nulirk eryles
2TanA}
Sn2A =26nAcost =
1+ tan2A
1- tanA
2.cos 2A= CosA -
S in?A =
1+
tanA
2 tanA
3tan2A=: 1- ton cotA-
H. Cot2A:-
2cot A

13. CProla 1t cos2A
cosA=
O Cos2A= 2cosA-|=) 2
Cos2A
2t- /- 2sin2A =>
S i n 2 =
c o s 2 4
2
Sin2A 3snA -HsnsA
4 Cos BA = cosA - 3 o s t
3tant-
tanA
S tan 3A:
I-3 tanA
2 an
ISinA 2sint eest . 1+tor
1-tan -2cos>t-i=l-25inA4
-tan
21+tan A
2 CsA=Cot"f -sin2= 2
2
2
2 ton
3. tanA =
-tan
1 8 72
36
Sin ho-25 VS+ o +2/FF
fotays o-2/S
CoS

14. Heights Distance
S i n Rule: a, b,c are Sides ABC a r angles.
A
Sint Sinb Sint
Casine eue
= /6+ c - 2be cosh
b la+e- 2aceosB
Ca4 B- 2ab eose
- n zule
m+n)coto : neotB- mcoth
- ) cote - cot -ncetp
m

15. Hemedial athamahes
G1eometoyfesmelats),
Co- rdinefe Sysfem:
Diefance
t6-
2-7 +(U. )
A B e
AB+ Bc = A A) ACteB = AB
o
BA Ae=Ba
(2) Co linear poin ts.
Triangles
Equila feral reangle "hzee side a g e egual.
ng tewrosides)
sesceles trangke: Two sides
a r e poaal C7
Reght angle triagle: heangl he tnanghe 9o°
Aceh Angle Tartangle : he argl re a a e
(ess han
C b h u s - rgh
raryl:
Kayks
aua
esMan (8o S m a e
han 9a
anded
Lko eeles trangle:
To sids ays
a s
egua
Gue tilatarals
Paraldogran: leoe Npesie
scds a 2 e paraled
> egecal
- Re ctagle leoo eppsik
s i d cese deagnls
e r e egual.
-Rhombus
hefour sides a
g a a .
cliagonel Ciie egual
- Sguas
f o r r s i d e s
-Area a triangt: | J J»)**2CyH)+3GA)
-
tHeea ef a uadailaferal

16. Sectden Fermelo
ma +nx
-Sntes naly
-Cober-naly /mx n my,-nd
m-n m-n
(
- Mid point
2
seeAon point: A,B to points A6dvetes sad 7
/2 &2;1|
centred efa trargle: ( , t+8, )
3 3
AA&.
Deaun centhea taiargl
hcentes -
atbtc
E C e t r enposit
tot =2, ( tD a + yt
-atbtc
- a t b t c
)
x-bat Cz
Ex tentoee eppesik o B-
a - b t - c
atb72-C3 ag,- )
C Cenhe cppesih to C . atb-c atb-e

17. 6iepmetey
Coodinas Seen
R H) be a ppinFin he coordinae plane
1
C
on z-azis f=°
-axis *
p o o n T
on z is
ef heJam Cz,)
Co)
tr
aris
Dis tance bleo too points PCX, J1) CX}
Cellinear poins!tso 2mepoints lles on a line
AB Bc = Ae A
*/siarges
Sum angles 4 :1Ro
Equila fesal tnange
a c e d s
a i egual
seceles taiangle hieo ides
a e
egaat
Kght engle triangle: One ange of A° I0
heuh angle riangle: hre es
tve actk Clens than 90).
Obtuse angle Taiangle one ange is obtuse (nse Mon90

18. Right angled dYsoceles rions t o sids a n
ght aagd A
Cuadrlatanald
onalelogteom: ftro oppsi/a es parelel h/
Kecfangle teeo onpesBdos &degonos cuas
Rhombus: four SiP
a egteal
Squas: four sices E two dagonals egual
) Cy) -G CI3)
ne sgre
m:n
Seehinfzmula ACZ1g) BC> )
m:D ne
s r
bnternal )
m+n m+N
(1ma2-ny
Cz ternaly m- m - n
cerrolla + dT)
micd pont
Tsesec hen points let A, B be two points. hen, the poirh
hch dirides
AB tn he aio 2 & 2:1 a z called pona
siseeion q AB3
Xf hreeorm ldee neet ataointhow y heloesar
C n i u e n k l r y f

19. Cenkrid e tiaag/
Median: The pasirg hrorgha veerr & midpoint7 pposies?d
oL As calted cdia
Centaodpoint cencuenen medians
G
,
S t» )
3
e Srecoe nas passtng & bcek a t s i t
S b e c o m e r .
Cincum phe: heLa biseeto he side ga s
SA SB=se
de ceohesftn'angle "he pontdcneurence qalktuules
r a e s
Ccly G, S, o hn heser poins a s
colinear alsoGdvtiks
h une sgrmens
Os Cos)
he ato 2:1.
|ncenteg rangh
Angular bbeces
1 line cohies böech th gven ary mcec&
y tero
nes Caltec gular
ksee foi of he tao lnes
In centa: The point coneusene emtma bseetzs o
angles g mangt
c a l i e d
à cenbe ofAk
ath2terz
Ak.
atbte

20. Locus
hepoh a po
hich moves in a
phne such hat
cSatsfies he geven cencleHon called a locs ot á curve
NoteO Y evey pont PC1}) in
abeu soatty h
cendikan C})= o then h eguaten hcas
, )
20
Noe-: aenesal a occs c a n bedescnbed hyk eguahên
pCxug) s a point t bces such hat oztby te=o
hen he egaatin et loces -P a tyte =o
Lf PO) apoint tin o ces such hak
o hen th eguain c locees
P ax +21t

21. STnaighr lines
Desanee - AB
Lne segment -
AB
Ro
AB
lnelinatioTn a line a line ma
ks a n agle
cwot 'e Cose<)|
evith
-
a x i s in
he r
d i % e c h o n ,
hen d
Caled
n c r a r e n
h e bne.
a
live,
hen ha t a b
lepe efa line: Y e s heinclnatien ofa line, hen h alun
he lne denotel
tane 'is talted
slapee
he bne , denoted
m
/m=fan Hey
slapes ef any
rwo
n a n -
Vesttzad
lines a k egual
hen
r e pasalleb hehe>
o o non- veetreas
Lns
azeLes
en the psodueto heirs i-L|
slepe4 -aris o
guaion gz-aa èy-o &y-azs
à x : o
gua h lne paralel
o a
z - a a i s ápassing hrea A
(1
. -ars
Slepe lire jointog he turopoirts A
-Gy) 8-Ca)

22. s angle blw tws no-Veshcal (nes
having slape
then
mm rzespeeive then
m-m
-/anp
tan O-t
1+mm
Nobe
s aak angle b boreshaviny slapes m, hen
-2L
tano + m,a
guaten stnagltlea in dereptfee
Slope-poin fosm eguatn t stagtbne passigg througl
d-H Cx-z)|
Tece pointforn: The eguohan staghtline pasi thrg
Stepe-aleteeptfpsan:
lntereeps 4 e lne
a line
intersech
z-das sy-axis
at Ao,o)|
Bco, b)epeteh
then, a,b ale
callead -inkceph
&y-irkseypt
eco,b)
ACa,o
Slope- ptercept/a7:
The eguatern f he lne lavy slye nE-vercept
is
mzt
Corolie
Eguain straighl lina
havwyg slyr m wih 1- inferept
-C--/mex-
slepe 9t lre axty te =o is

23. Slope =
Nofe
Slope anyline pasallel
o he tne arthytc#o a
Ler
quaian of any
Cene paraltel
o the given lne
a t y t c = 0
eguefon
given line aztyte=o
lar+by+k=®7
Ler
babr-aytk=o
ntercet form he eauafon of he line havirg -ntrcph at
inkreep bato) 3
Cenolery:
he intrcept e he bne
a aNby+e= o ato, bzto
-intercpt=* - nkreph
reo triorg fomed he bine aztbyte =o
i 2labl
epepdieaular JarnalNcomalfon
he eguaion et h. lin ahich cs at a disfanee of Pf>om he
gin &
ies hfo oto360" Ce s S 36o) heangl
maete b he le7 cwith
dièecton o - ars
/Tces
.cind .
P Normal frm.

24. The cguahon q the line ar ty*e sO to Perprdaaar fom
a+
- a -b
>0.
a+6
Coolles' he pependicaar
disfone- rerm he point Plx) to
he line artby te =
a+by,+
|
Note he Ler dis tence feem he point Co,o) to the lne axtbytcEo
a +b
he acstanea blo hro parollel lines ax+byte,a+y
a2 +6
ymeteie fonn
The equaon
lne passi hrogh Ct)2 hovn
nclhina frbn
- 4-8 e oe(o )
Cas sino

25. Farometale fcm:
PCx, ) ang point un he lne passiy hregh He point
CH) hauiny
ineli'notn
Men L -2,trcs 4A trsns
chse l u he isfance frm
Ato P
he sato n hees he line- an thy t e = o
diwel halu
Jrentjoining ACSJ) B: CJ) n*b2
uhela
u a x +
+e
Noe:-aze
diides
he lne segmeno
hpoit
Nor -ars ivedes he ine seqmentfoning thepou
AC , B CIy)
tropoines ACx) E C
S a m e
Side. ine
baa have p o s i gn
on
lotezsecHon duro lines
two lnes whih aseno
pasated
hen hy iressecteach
o h r a t a p o t ,
then hap
ontà
Calec point o
nfessechin f hso lines
a thteo
ab-b>4, ~o
bv