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65 views47 pagesUnit - 1 (Business Statistics)
Study purposes
Uploaded by
sharmilaalagesan47Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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/ 47
fa Matrix
an @irangemen;
3. "The numberg
or double bars.
of ele ame,
are
{rices-
all ma)
salesman has the follow
y three items A, B and ¢ Fecord ay
ing sales
; mit eo ae
90: 100 20
Jon ee 130 50 40
fe E 60. 100 30
Mar ike written in a matrix form as
90 100 20|
130 50 40
60 100 ful
portance
oa
ag watris enables compact presentation and fac
manipulations, it is used in many fields of study
ent for computer operations also
The common operations of addition,
: i - esible and
snsposition, Inversion, ete, are possible and
sigebra 886
e from code message
oes tatiatics Gn Design op
ete), in eonennies Gh Sock
- Gommerce (in Lines
a
etc),
‘etc, ete. the matri
a ) HEX form
ge an important
solution of
ariate
Sutput Tables:
of Expens?
Experiment
Accounting, Input-C
Programming, Allecattos
ia the most convenient one
3. Notation
ABC, _ are used to denote matrices
_ with two subscripts denote the elements"
ea the row while the second indicate,
8
Capital
Small letters a,b,c,
The first subscript indicat
the column in which the element appests-
The general form of a matrix is
a
au re a
a 402 3 5
A : a,
i sg : a
ai, ee ars
nt in the first row and seconq jt i5
second it iS
is the eleme
For example, 4),
e second element in the first raw.
column. That is, @,9.18 th
In 2 more concise form, the above matrix is written g:
as
A=(,) mx"
.. is the element at ith row and j th column i
réferred as (i,j) th element. ae
are
4, Order of a Matrix by
Order of a matrix indicates the’ nw
> mber of rows
number of columns of the matrix. The general form ee he
is of order ‘m by n’ or ‘m x n’. The following matrix is a aa
order
2x 2.
[s 10 19]
41 49 50] * and
aSeciai |i” _ yare Matrix. wi., re
Linear 1. .olumns of a Matrix gn
UM ey
3 8ong patie
pet“ nateix. I the Matrig 24 the
nu 8 Called
number of rows —
jp the
I
le mba
alled a square matrix of Orde,
¢
1. i8 a
trices, or he a ‘ é
ments, re example, Eve 15 3a Square
licates f
now matrix. If there jg nly one
a
a a row matrix or a row Vector
hs
lle ee [10, 32, 50]
psa Vo
ie of order 1x3.
It 15
Column Matrix. If there is Only one eoly~
3, Co
Mn in a Matrix,
! da column matrix or a column Vector,
valle!
cond ; i © = [ ‘|
: gxample TA 10.
! 2xl.
as) 1. is of order
ieroin Null Matrix If all the
4, Ze
elements of a Matri,
it is called a zero or a null
i
matnx and is denoted
jy 12005,
no o|
Example 1. 0 = ° v
It is of order 2x2,
Mio.0 °
Example 2.0 =o 9 0
It is of order 2x3. a8
Tivo matric
d only if
. Equal Matrices
at is, AB) if
equal
ae A= @)man and BS
@ they have same order, that is, A= @ man and Bb, dmxn
and
(i) the elements at the corresponding places are equal, thay
is, a, = b, for every 7 and J-
ub U
[2
Example 1. If A=|5 7
ZT
Example 2. If A = [3
\o
mplies x = 10 and y = 14
6. Equivalent Matrices. ‘Two matrices A and B of the
tame order are said to be equivalent if one of them can be
tained from the other by elementary transformations,
It ig written as A ~ B and yead A equivalent to B.
Deseription of elementary transformations of a matrix ang
pple for equivalent matrices are given Tater.
% Diagonal Matrix. of whose
lements. except those.in the princip ..t or leading or main
jagonal, are zeros is called a diagonal matrix.
A square matrix all
a 0 0 aS 9
ie ae ore a 0
a Gig Ag anaes 6
‘he matrix: A =
0 0 nea
a diagonal matrix. It is written also as ™”
A= tag. (4,, aj - - - 4,,)
22 ny
That is, A is a square matrix such that aij = 0 when vy
je"
ab is, A = diag. (5
t is,
i) are ae
is
ls Matri>,
a. Peper ai Zonal ay, Yong)
in the agonal 4...‘ A
bmxn - ents al 4,
yor OF
| Maes
1 that pe mala = 16 2 3
phe 4
0 0 =
atrix. It is Writte: fee |
ee ata 7 also a. a
ee cine Ga es 8, eal
[ef
nat is, Ais a Square Matrix auch ¢
F the is 0 when 1 + j, geet 2
be yt Bi
8 When is
is to be seen later that A= a Tay
{ is
here | Sam Mat
9, Unit Matrix or Tdentity Matrix A ei a
diagonal elements are Tanity) each and gon to
and e are Zeros is called a unit Matrix oy an ee ae
: om" anor by the alphabet L In’ at r Conta i en
eS in which all the elements Mm the pris nal a
x
is called a unit matrix, A Unit Matrix
i matrix. The order of the Unit may
fa ne of the alphabet 1.
she §
el
ACIDal diagonal are
1S 8 partion
trix May be
ay. Case
Written
Mendy =|,
Heneé, Ty = jg | and I,
It is tobe noted that r
4 = 1 for all i = j
0 for all i = j
is a square Matrix such that
as
10. Symmetric Matrix, A square matrix such that
a. = a. for all i and j
Wy iL
called a symmetric matrix. R90 1c
9c
spose of Aanditha, |)
A where A! is the t
That is, A t
been defined under “Matrax Operations 1
5 7 9 n°
Example : A=/7 © 3] a
9 -1} Bo
LU. Skew Symmetric Matrix. 4 square matrix such thay
ev
ay = - 3 for allt and j be
i
is called a skew symmetric mattis
That is, A’ =
|
Example : A = -3) j
| a} pee
I er
Fete ‘ni
12. Triangular Matrix. jangular matrices are of two
kinds, viz. upper triangular matri* and lower triangular matrix
A matrix A = (Quan 18 38 UPPSF triangular matrix am
A ii’mxn a
40 for i > j. is
i: 5 wer triangular matrix aAgere
A matrix A = @penxn as a lo ie if a0 en
for i Two mateo. nS
ni conformable ‘for additi
0 we and only if they are of the eax
gsitions are to be added.
£0)
Stern,
pample Bas {5 ks ei and B =
‘4 Bae Ee ae oe =
i 2
ular Bae A es bre | =| : 1a)
- if 4 properties of addition are given in subsection 5,
ven (i) Subtraction. If A = (a) and B = &,)
self. . Two matrices of the same order
52 (aj) man
formable for subtraction. That is, subtraction is defined,
pa only if matrices’ are of the same order, Elen
ind
ee cal positions are to be subtracted.
ew “1
=|5
[es
| and B
1 3}
Set Reso |
; (B.Com. Bharathiar, Nov,
Example
find A+B and A-3.
iy JI
Ss ‘on: A+B = cae 8\
Solution: A+B er n 5|
ay is a real number in the
To get a gcalar multiple of a matrix
is tm be multiplied by the sealar, jp
their product 1s KA which is
Multiplication. Seal
2. Sealar
tions
context of matrix opera’
the matris
dA is a matrix,
every element of
K is a scalar ant
l 1
a scalar multiple of A-
0 4] o 2 ) @ -1
For example, if A42 6|+ 345 6 15| and — 4A=|-8 -20|
[7 9 -21 27 |28 -36) 3
235 [s 1 2|
Example 4: If A= |4 7 || and B = \ 2 5), show
1 6 4J 6 -2 7]
(MLA. MK, 492)
that 5(A+B) = 5A+5B.
A and B,
Solution: From th given matrices
ip 4 7 25 20
A+B = i 9 14[ and 5(A+B) = \a0 45 70)...
7411) 35 20 55] ‘i
10 15 25] 15 fogee
5A =|20 35 45 5B =|20 120 25] and
5 30 201 [30 -10 35
[25 20 35
5A+5B = /40 45 70 : iat
I35 20 55 ee B93
Pee): SCA...
aaoele IEA
od B WBE 2A45B
From, .
sons" [zo Ble acer
| 5B = lio as} and Dre
f
., [26 1045b
CyEReS, faa 4542
6 a
4 A445
a (1) and a
45.
meieanid =. from fe 2
fr “hb =
GFA
B ample
“pat AATSB+2K = 0
Mt
(M.A py
‘dasan,
ee «on: From the
poe. jutiow Biven matrices
matrix,
alar. if
nich is
A and B, as,
eel
1
16]
20|
36]
show
AQ2)
|. X may also be determined ag X = 5 GA+5 By
QM
©”, Maltiplieation: If A = ( ip ° nd B=) their
vq AB is a matrix C = (c,),.. where ae
cc -
oj = Fil by + 4i2 boy + ee a + a. bys for all i and j
That is, ©; of C is the sum of the products of the pairs of
2 nents of i th row of A and j th column of B. The elements ary column of B are py
to be written in th,
This is possible only
is equal to the number of
AB is defined or A ig
if and only if the number
of rows of B. It i to
B for multiplication, B
tion.Even when B ig
d not be equal to AB
2 matrix:
Hen
to B for multipl
number
formable to
for multipl
ation, BA 2
been post multiplied
a by A pad
ye been pre mull
prgdvict of A by itself ie
ware mati
a2 Aa ailarlys AS =) Aa ee
Subsection 5 deals with the properties of multiplication a8
also. he '
3
Example 7: If A = (3.6 ©) and B= 1
ix Bul \el
AB = (344+ 51+ 6x2) = (29).
al
Note : Multiplication is by row of the first matrix and 9,
column of th ond matrix. That 18,
47
Example 8: If A =(2 35) and B =]-0 4
a*% 3x2 |6 9
AB = (244 +304 5x-6 2x7 + 3«1+59) = (-22 62)
1x2
Note: In AB _ , there are 3 products in each element
t43 3x2
A=|s | and B=
24
Example 9: Hixamine whether AB = B and BA = A, given
[ fi (M.A. MLK. A92)
8 on
id.
qalrices
_|2xd+axs
pa = |ax4+Gx3
:In A
Nets 2x2 Be
just jn each element
geample 10: If ax 12 3)
4g '
3G g| an
AB. aE
a
olution:
fyecde2eo24+8*1 1x o.5
-1
MP [sno Sel 8x-246,_
fone
o0°o
9090
Note: If BA is needed, it is re i
as
follows:
[-yxi- 22-43 =1x2-8x4- 4x6 _)
|, 4-1 eee Fx 2A ode dug ye kt Axa]
ee [ixd42x2e4x3 Lxht2xdiing a Ba ae
(47 -34 51 ‘ a
“17 -34 -51| and BA + AB
li7 34 51
Example ' Ti: Write the products AB and BA of two
Went 2
{1}
al
(B.B.A. Bidasan, A99)
Solution: From the given A and B,
1x4 4x) On i
896 i
.
a
¢
ay it
aB = [1xl+2* < ee
eh They ie and ied a4
fect az 2a 2x4 a
‘= faxt Gxt Sed Sed ed
[ana a2 aud 2
element
2 4 products in th
B , there §
Note: In A
1x4 4
ch ent.
there is one product in e a
| i!
2 0/)8 4
tue 4 om
BS 2 1j)14
_ he
(B.Com, Bharathiar, N95)
fa 2 ol fa tee 8) _ rt
Solution: [1 2 9] {1 <2 3):/! = (27 6 9] |6| = [2am
7 21jl4 4] p
n r
multiplying the first and the second m.
4d :
Notes It is also equal to [1 2 3] FE = [288], multiplying
the second and the third macrices first.
13: Two shops A and B have in stock the
Example
following types of radios:
Single Band Two Band Three Band
Shop A 23 20 15
Shop B 40 10 8
Shop A places order for 40 single band, 40 two band and
90 three band radios whereas shop B orders 26, 30, 20 numbers
of the three varieties. Due to various factors, they re
half of the order as supplied by the manufacturers
the three types of the radios are Rs.100, Rs.220 and Rs.300
respectively. Represent the following as matrices:
(i) the initial stock
(i) the order Bo
2 supply
anal stock
) i of individual iton, :
aa cost of stock 5.) tothe
tho
a
i!
apiution*
tial stock matrix,» me
he :
lement, ? ,. oft ee Ba
2
mie 8 d
1" ply matrix, a
s
qe
‘le gnal stock matrix,
the
ped vector, They 2anl
Moa) i . L309]
N95) a jotal cost of stock in the shops
he 8,
ie =} | 100]
ee ire EB 25 1s} |220) = {200001
Re in i300 Las2oq|
t of stock in sh 7?
total cos! SOE AE Sipe yn
dying La = Rs.16,200. Rs.20.200 While thay
* oP
- se. Let A be a matrix
; Trane , to OF order .
roel me DY KOC AN or oven Sinem
ma 4s obtained by writing the first roy of A on ae
fc the second row of A as the secong Pe
fA as the last column. This
jrott © n of A as the first row of
wm: 5
s “oe And the
Process in turn, gives the
AY, the Second column of
fi a second row of A’, ..: . and the last column of 4 as
and " | TOW of Ar i
bers * i |
only le 74: If, A = a ‘ At ne : x
Oe csp 2 134
300
Note : (A’)’ = A 7 nT
An? y
hen
Example
(B.Com, Bharathiar, Apgy :
Tal
‘ | and B
5 1 a] and E
From the given. matrices A and B,
Solution:
: 0 | i | cy 2 Ar iB » Oy
ap, a(t «| st OP i al &
», BTAT = GAB)"
From (1) and ©
ja alao dealt with in the nox
fhe properties of transpose
~ibsection (iy
6. Properties.
(a) Addition of Matrices.
(i) Commutative. if A and B are matrices of the same ordey,
= Bra
(
AtB
{ A,B and © are matrices of the same order,
‘A+B)+C= A+(B+C)
pect to scalar.
(ii) Associative I
(iii) Distributive with res
k(A+B) = kA+KB
(iv) Existence of identity. The null matrix O of the order
of A is the additive identity of A. It exists and
AtO = A = OFA
(v) Existence of inverse. —A is the additive inverse of A
At(-A) = 0 = CAA
(x) Cancellation law. If A,B and C are matrices of the sam
order, Att = B+C implies A=B , afultiptioation ,
commutative
¢ have bec
AB «= BA
when
, ASS)
yiiplication is
Fi x Pista
ow. 1 AB and c¢ 0 “iy
» respectively AGC,
B and © are py
(a+B)C = ACHE
sveb May,
renee of identity. }; 4 ot np
next
rder, _cstence of inverse.
° non singular, : ni Pht a
, such that are matrix § ,
AB=I!I = BA
rder,
Similarly, A
Mer Note ; 1. Phere exists no inverse for q
2. In fact, B = A‘ and A= Bp
Rp
‘.
B=0 or both
ime
Consider A = 900
a null matrix:
is
nor E
xn, the null matrix O of order nx,
= of order
PORE TG and OA= O,,,,
pec
(viii) AB = AG docs not UBPIY
(c) Transpose-
age of the transpose Of ® matrix is the oviging
matrix itself
That is,
‘The transpose of the SU
he individual
= A+ B
@) The transp'
ay =A
mm of matrices is the
: Sarasa
| matrices. \
transposes of t
That is, (A+BY’
is a scalar.
“i
i) GLA)! = KA" where Kk
¢ product of matrices conformable
tion is equal te the product of the transpose for
nal matrices taken in reverse order. w
Cy = CBIA,.-...--
(iv) The transpose of the
mnultiplic:
the indiv %
(AB) = Bia, (AB
7. A System of Linear Equations
Consider the equations a,x + Ee and
AgX ot boy ¢,
2
. are linear as the index (or power) of each unk
nown
Th
¢ well as y) is unity(1) and there is no praduct ¢,
erm
(variable x ai
of the unknowns.
By wri ing the coefficients of x in the first colum
eae of y in the second column, the ee
" tmatrix
the
]
1) ig obtained. A is called the coefficient matrix. [
- The
unknow!
[x| ;
|*}. The constants in the right hand sides of the equati
ations
3 are taken in a column and are denoted by X wh
where +3 °F nxm ade =
nc? the above system ay
re As
a : Qua
le for f which pale coefficient mma
Ses of me yector and C is the constants + ee
) (
! system of three ]j
gor the a4 inear equatio,
yo?
] ayxtbyytey2 = _
a,xtbgy tcp? mace
/ agxthaytege = ds.
TOW pattie form i = & where
; c y ry
term [1 an + (2h)
ye |4s Pepe = |¥| and c =\a,\
ae 45 by %] i \a,|
an 3\
atrix 4 systema of linear equations is said to be ee
The: , 9 solution. On the contrary, a system cf linear equatior
ii to be consistent if it has at least one solution, A ht
, of the unknowns 18 a solution (set), if it satisfies ea
ry equation of the system. A system has a unique
jA|# 0. That is, A is non-singular.
here
Lions
I
Ei
jrion if =
02 F
fu is
a aystem of Linear equatig,
; sous equactenaeme”
QO, a null vecto
of linear
on a
homage
ro
When ©
je called a system
example, cae
eb ae a eth ytey
a,xtbyy
ayxtbyy =
are two systems ;
1 equations, AX=C, is caljag |
called
equations.
stem of line
neow
a ey!
non-homoge
When C # 0,
a system of linear =
refer to Section 11. ;
a
For some more aspects,
8. Determinants
ix A we can associate a determinans
bol |Al or det. A or A. Whey A
is a square matri of order Nn, the corresponding determinany
JAl, is said to be a determinant of order n. A matrix is juss
an arrangement and has no numerical value. A determinant haa
numerical value. Further,
5 5 96 1 6| 7
[2 31. fe 3] ' | s| eral F | are different matrices but
6 9 le 1’ L6
the corresponding determinants have the same value (-21). Ip
enclosed by brackets or parentheses oy
mbers are enclosed by a pair of
With each square matt!
which is denoted by the sy
matrices, numbers are
double bars. In determinants, nw
vertical lines (bars).
Differences between matrices and determinants are a)
6
follows:
Matrices Determinants
1. Number of rows and
number of columns
are equal.
1. Number of rows and
number of columns
can be equal or unequal.
2, Elements are enclosed
by brackets or parenthese
or double bars.
Elements are enclosed
by a pair of vertical
lines (bars). jl are arrange.
si By interchanging
Wations){*
it rE
F at® Jenta in a Mat
aS. For. 1!” gem
rix
fo (matrix is obtained ee
bs
Ro first ord
ie eee Ordey 4 e
LP alte of term inan iat
be be é, Hy ~ he. mp
we jel = 5) 10] = 0; I-13, ee tle
called” ..f- feeconad : 13 :
jalue of a secon Order eternina ny
abe
1
{ ne a mee Of ad _
|
ig
linan{ uy =
i 6x3—-4x1 = j4
hen A le |
inant |4
S just “9 Ee(-6) = _9
at hay | § Fil ca)
37 :
| alue of a third order determinant
ne Vi
8 bu, fh? :
are considered and bg
D). Tr io methods got them ‘eve, tye
ESO) sue
air oF d i: The value can be found by e
iets any one row or any one column
i ae is multiplied by the value of the second,
foi OF 2 t obtained by deleting the Tow atid the column
— eterminan! tf is, with the sign as such or with the
kd the e. Pe the row number and the column number
a. (f the even number, keep the sign of the second
: dement is a such. If the total of the row number and
Se vcmbor of the element is an odd number, change
Shlwun num
termi y to minor and
: of the second order deter minant.) Refer to mit
an
‘Panding the
Rach element
change ~
904
cofactor and also examples below The total of all such product,
is the value of the de'
dn short, the value of a determinant Ne the sum 6f the
producta of the glements of any opener omc O ag and thei,
we cofactors)
respectivi
the value of
a
ne used for finding
‘This method can |
f any order other than i
determinant ©
at a | 5
i al ape] =a Paya) |aeeaate| 1g a
| a1 “22 28 11 | 250 a,,| 1 a1 al \Pa. 8
' a1 Aap
J45, a2 729 |
(Or
233] wal 4 ag GC) la a Oe en ol 3
[332 433} a2 89 22 F2al i
(Or)
a, a a a. a
1z 713) 1 713] 4 t rst
ag) + agp la 3 agg) (Or
21 ie aa 22 |a,, 443 en ee.
is taken. ;
Usually, first row
the value of the determinan:
: Example- 16: Find
{3 -2 il
Pee (.C.W.A. Foundati
I 2 i| ‘ion, J99)
Solution: Expansion by the first row:
Pid.spajookaokd
erie Pee {tat aa
= 3 (4) -2 (-3) + 1 (-l) = 17
Example 17: Fi u f i
ae Find the value of the determina
Reis. 3, 0 .
. a (M.A.B'dasan, N!
Solution: Expansion by the third column: 05
2 (Sarr ;
oto * Bograny
gnding the value of la
the so Vay. \"a a
heir
fa et
ir a
we
maz] | *! a)
Rae jonsidered. The produc :)
aan fqn ave to be addeg
2
AY agg 921
1
and ,
eraiete © be Subtract
me mae.
2 83 [92 “33 12 Ogg a,
i! esl 22,4 i
BaP TP value
! {
si
ant Bey Oo
‘ ae ’
Ge tion : Consider 2S
gol petlaoa SS
)
al
“1
|}, Prope:
ad ‘
value of @ determinant remains
47
We = 3 + 84 + 28 - 49 - 18-3 = 4p
724
rties of Determinants.
OM ianged tuto columns (and then colunns
" This means |Al = \A' (or column) of a determinang,
ts of another row (or column),
not change.
|
wy 4a] +a |
Pare ‘s
the wh element of a an
pe of WO elemen,, ©" coy
the 4
sunt of two deterin;,, =
nan
are
ver? element of a row (@
yA "determinant is zero, Solunn) j,
af es 1
edie 12 a
by; °8 {0 O Ghar > U
ab 22|
° cramer’s Rule Or Determinant M
Kay cyamet's ‘g rule [named in honour of Sy
Fy (1104: 1752) is based on de
Kay: aystem of linear simultaneous equation
Wiss
terminar
18,
ret the system of linear simultaneous
€quations to he
pital be
pire ax + by te C2
ayk + buy . + Coa =
a,x + bay + ¢,2 = 4,
jin matrix form, it is AX = C where
~
to
I
ao 908
Hi -
a, b, = =
reise ao=}%
= ye ant
A= [4s Bs “|: x fy a
3
. la the cons
i in A is replaced by tants
x coefficients column in coefficients column ay,
Similarly, 7
column vector to ect Ay em
«
- col zB
coefficients column in A 32° replaced by the constants colism
vector to get A, and 4, respectively.
ae tae avimec ea
I a be
a,b, %| ria iis |
eta pepeal ee hence and A, = \3,° % 4d,
beh aha he a) Bae dees aoe
d, by: &| a? a oy
pA! 1A. |
"
3 eres
By Cramer’s rule, x = jal ’ ¥ TAT and z=
For the system of linear simultaneous equations
a xtbyy = % and
agxtbay = Cy,
:
b c
pores (ies cen
ay Ps! *
ea
] I,
comb | : a,c
A =p? J band A a :
x £5, b,| ¥ a,
é [Al |A |
By Cramer’s rule, x = TE and y = ke
Note: Cramer's rule is also written as
4. ae A
x= jy=sqiandz= =
or
ny es
x= of and y=
as ‘the case may be where
A= /Al,4 = /A = =) |
eee AT A, IAI and A = 1A. 309
examen 1° Sclve 4,
fo},
‘ “Mowing nee
3x + 2y 5 8 ‘ions )
ve 6x — By = 5
gomeciven Mations ; ay A.
fioR me ne ,
soltt fl : C Natriy ee a
e 5 > X= |x
ants (©) a [5 ~3 Ly] And ¢ ey
1and z ae |
fi eee
co}umn Et 2 | iaaet ¢ ee ae
$ a j cies
fe 2 ci
a, (ae= i =| i (5, ces
a,|” Es
a...
2 zy Ault 5,7 Bad
2 Vag Ly 1A
Cramer's rule, x = eT oe
_ IAs
ea = aS
! fication: The correctne
if
Ver
88 of the val
1 as follows:
ified
be ver
att
hes fj
Und for x ana
f x and y are to be in the left
The Mer Nar both the equations ang Simplified, ELH.
side (LBS. ide) for both the o,
poe aright had s
? iB
pHs. (4
uations, the Values Are
there i © mistake
i Te 18 some Mist
WISE,
| at, Other
pect.
Substituted
le, L.H.S. of eqn.(1) = 3, 242 158 RAS.
MMS of can)
= Ox 2-3 lets RES.
I = 1 are correct.
ti = 2 any ¥
20: Use determinants and solve
| Example :
i
a
3
a
=4
Sle ol 910
1
ere:
ee eetiavet seer eae
ne,
Pie piven eqnanons beccn
See |
ax) - 2 toe fs and © = (4|
oes , = C where 4 = (3 -t]° + lsh
ie, AX t
2) 1x (-1)-3* 2 ea
lal | me '
aes Bo ge E1)-Be 2 = I
A, 1
‘
alee vt
1A, | cee
by Cramer's rule,
i
ines Zora
Verification:
In eqn), LHS. = B+} = 2% 7 ee
Fee Ch de aaeize geo “loge es
Hence, the solution is correct.
Note: The unknowns may be x, and X, or a and b instead
of x and y.
Example
equations by Cramer’s Rule:
2x + By + 32 = 22
x= y+ Z = 4
21: Solve the following system of simultaneous
dx ¢2y-020.=°9 (.C.W.A. Inter. D98) Ba ification:
r 68 4 83,
1 = pe
. (), DHS. = 2x55 43x 5+
| a8 63 , 119
heqn (2), DHS = oo oar
ee et 3
van. (Q), LHS. = 4x37 +2x oe
8) °, the solution is correct.
lig
31 912
the case of matrices, th,
ix and those’ of every,
Jed in pats and thei,
the product matriy
4, Product of Determinants: In f
of every row of the first ma
econd malix are aay
propriate ace
f the first determinant ang
mant are multiplied in
wo!
propriate place to ge,
slements
column of the 5
sum ig written in the aPP
rminants,
ry ro
secon
In the case of dete!
vow. The elements of eve.
those of every, row of the
sum is written
When
we of the
ace is Ihe
d determi
he ap
pairs and their im the AP
the product determinant. the ih ae Ae the first
determinant and the } th ra Re a e aan are
multiplied, the appropriate pl that 36, ] th element in
the i th row.
ALJ BI is defined ov
multiplication if and only if |A| and
Te may be noted that 1A] and 1Bt
be faund brrespec'
product of their values can
ia conformable to |B| top
[Bl are of the same order
can'be evaluated and the I
ive of their ord. iE
le
{Al
Remarks:
1. The value of the product of determinants obtained by
each of the four methoda ©!
Jumn or column by row or ©
e second matrix can be multiplied
f the first matrix or every colimn
pro}
ff multiplication, viz., row by row oy
olunn by column is the
row by col
same. Hence, every row of thi
by every row or every column 0.
be multiplied by every row or every
of the second matrix can
coftumn of the first mratrix.
2 Determinants of different orders can also be multiplied
after writing them as determinants of the same order. e.g. if one
i «, ja b
is of order 3 and the other 15 |) | (of order 2),
determinant
la | : 100
aj can be taken as |0 a b
joe d
22: Evaluat Bed 2 ‘ }
Os ttle ce
Example
(.C.W.A. Inter. 399) a
value of 5 9
example 28: Fina
solution: Required p,,
1 C243 x0 at
9 oxC2p+(4) x0 oat
LEDERER +9xO 5x :
Note: Its value = - 4128
5, Minor and Cofactor
Minor of an element
er order determinant o
jum which contain the el
genoted by M-
Depending on the posit
is the cofactor or the minor with ct
py the element at i,j th position, (-1)”
if the row number (i) p i
+j), is an even numbe
is an odd number, -
is denoted by Ay
ence,
jement, @
When (+)
fan element a; 914
an even number
add number
3 M, if Hy 3s
A, -M;; if Hi ie an
together with: thebr minors
fhe clements of "1
‘The clements of |,” b,
cofactors are found below:
Cofactor
Element Minor a
a, iby! b, ae
By lagl ae
ay Ib = eal
E = a
by lal = 8 i
The minors of the first and the last elements are cofactors
also, For others, the sign is changed
Similarly, the elements of together with ther
minors and cofactors are found belo
Cofactor
Minor
= bc, — bgt,
= —(agcg~ 49)
= Bybg- Sghy and
ir
a
eee 2b, a,b 248.)
la, by 1727 Agb, 6 1
4 a7 a by .
wine to the def. Inition th, a
rely HOt changed and chay,,, "8" af
08 Bed to pet
no Bet th
te
aa Beg. 2 tm
an and cofsctorg
on of |5 0 TS of al) eh,
golution:
Element Minor a
3 "i etactox
2 5 be
5 9
0 : =
3 5
snors and cofag!
pxam Je 25: Find the m
Bean
le
I the
ements of he
9
Solution:
flement Minor Cofactor
5 a = boas) =k
6 2 ™ = 0x9-(-2)x(-3) = -6 6
Fl Y : Broeeo =| = 2 2
(“0 we = 26 ~ 2g
0 ees| Gx9—4x7 916
1 ES 4 gaat = oe
-3 bg ‘ Be C2 Gn ued at
-2 iH Ee Bepresyetn 7 aan, =
a le 4 oe a ae
ee.
of a determinant is the sum of the
row or column and their
the elements of the
26, 59 and — 32.
Note: As the value
products of the elements of any one
respective cofactors, consider, for example,
second row 0,1 and — 3 and their cofactors ~
The value of the determinant 18
Ox(~26)+1x59+(-3)x-32) = 158
e third column 7, —3 and 9 and
From the elements of th
f the determinant
their cofactors 2, —82 and 5 also the value 0!
is 7Tx2+(—3)x(-32)+9x0 = 155
he cofactor matrix of a square matrix
d it is obtained, from A when every
factor. Therefore,
Cofactor Matrix. T
A is denoted by Cof A an
element of A is replaced by its co:
A, An Ag] artis ee
CofA =|4,, Age Aza] when A =|2 422 “2s
[Asi Ag As 43, 832 #33]
Adjoint Matrix. The adjoint matrix of a square matrix A
is denoted by adj A. It is obtained from A after writing the
transpose of A (ie.A') first and then replacing every element in
A‘ by its cofactor. Adjoint of A is the transpose of the cofactor
matrix of A also. Therefore,
int : An 4,; {
An A, |
A= 2 When
Aa fe Aas AS | fa
one of the inportane Propes
A. (adj A) = | 4, ie
pis property helps to kno
Ee from 8 given matrix 3"
af another property is adj(Ap) - es
ound
~ fadj By,
pxample 26: Fing the adjoint of A. | 1 sl
1 -3| :
olution: 2° = 15 0) 8n4 nai 4
yote: 1. The elements x, Tie Principal
yi ged while the signs of ot
cham
al diag,
thers nal aye
4 Ts have ‘changed from .
ii = 1x0-(3)x 5 = 45,
j 2. [Al ~ a an eli a}
(adj A). A = A. (adj a) - ale 1 yo i Stau
ia adj A has been foung Correctly,
ence,
(2 -1)
= |2 i how tha =
Example 27: If A (° Es Show that A.(Adj A)= \AML,
4 (L.C.W.A. tnter es
0
4
8
4
2.9 {-20 8 ‘\
fos A'=|0 4 Picky r= 106, sel
Solution: / ee \-20 16 3}
ee a S871 20) = -20 ‘)
FAL = a. 20 8 4)
a(Adj A) = |? ‘| ; iy cofyetor GH) adjoint anc s
31
Example 28: Find (i) nuine
Gv) determinant of the matrix 4
ne
Solution:
-1,-1|
' 1 6
(i) Replacing every element by its at 5 13 |
1 =]
sof: 1-5
(ii) (Minors with appropriate signs) Cof ae
r
fet a z|
(ii) adjoint =| 1 1.-5
|-1 -5 13]
|
31 2| =
Gv) [2 6 3} 3x(-ItIx1+2x(-1) = —4
{1 21
9, Matrix Operations - IT
x, The inverse of a square matrix
1. Inverse of a Matri
exists if and only if A
‘A je denoted by Av! A“ is unique. A!
© coneiieutan ie) (Aleo. Fusther, A 09> A:
the inverse is the original matrix itself
1€., inverse of
Two methods of finding the inverse are considered after
the remarks,
Remarks: 1. A rectangular matrix (number of rows #
number of columns) does not possess an inverse
Peat gee ree
This property helps to know. whether the inverse matrix
has been found correctly from a given matri S19
pA and B are equ.
I
® Matrices oe
3 4-1 exist, then a
oi ® a,
Be pyi=B).A
a
phe following two Methods
a
Eive
vis tA ads a
4 gethod vz ry a
“ A
, element in adj 4 18 to be divides by
pvery the inverse matrix of 4 :
get |
i 29: Find the Miverse of |a bl
' ele le al
-pxa =
: BBs Badasan,
ab =
jution: Let A = al os
Sol
P - la al : ye
| lees |. a | and Ale
A =lb
30: Find the inve
Tse of A = |%
gxample
Ay en} and
[A*1= 0.25, (A) =| o50 a0
-ly lea as
9, (A)
0
0.50 0.50)
soe 1.25 matrix
the
2 he inverse 7%
xample qeeal: ahiodae! a
fi ae s GBM. Bharathiar, N96)
Aaa 4 5
af 6 -7/
fenes o]
Solution: a =| 0 4 —4]
aa Cofactor
Element Minor
| 2 2
: a | = 28 + 30 2
6 -7
4 or eeram artes = one 6
# la Hi :
Op eonio eats
o Bio
auto| eee oes 21
eo Ip 7 a
Mee O| 02 oe oo are, -1
ie jaa -7 ;
: eas Onan ;
= s 2 Sis 077 = 18 a8
eral
: i oigtio* 2 6 :
deta eh ri
al = 4-0
Je 4l
Based on the first row, |A|
u
1x2+0x21+CD(-18)= 20
2
eee di a : nal 226 aid
adj A 5 21 -7 -6|; A= 77 (adi A) = 35 | 21 ae
[18 6 4] [-18 6 4
Note: A. A'=I=A71. A. Hence, Av! is correct
Method 2; Finding A7! from an equation aA+bl =
“?+pA+cl=0 or ad*+bA“tcAtdl = 0 or ....
0 of given equation
re
rand
$2: If 10a,
example
ution: Consider
So.
ah-50T. AT = 0,
A
- 507
jo “1
-50A7
=0
= 101
al
A
note: AA =I=4
pxample 38: Show
ation # «
ae
(C.A Inter,
Solution:
22
we AA
(9 8
8
< Sopeis 9 8) - |
peda ls 8 9)
Pos multiplying both
Me Pgs - ola
A-4I - BAT!
- 65 ia!
A?
eat
Len
92)
© be
substituting for » Aa
> jigs: Sto,
BOY .
104.
2_4A-5I=O.where 1 is the ;
50 matrix. Hence find the
. at
(1 22 1
21 2! (2 1 al=\g 9 8|
meet) N22 1)
\B a 9)
8
Is a 4
sides of ae ~4A-51 = 0 by A!
rae
0
- (A-41) and
1
3 (A-4])
Tew,
“8010, ; it
At
that A x
entity my
inverse o¢ Is
N17 ang Lowa Inter
* and 0 den: he
r. D9¢)
2 2\ \9 8 8)
1
i act ia corres
faeeceen %
Note: ATLAsEAAT ane hence
1 e matrix & Teen
that Us Be
Example Sd: Show : '
a gpd + oa te Hence deduce ore
CA Later 3 and Foundation, M95;
(Ce B.Com. Bharathiar,A98)
wl
r
2) -21 30 -30 |
Post multiplying 4’
AS AL ¢A2A9A.AT ALA” = 0.47
. AZ GAHgI-4 A= 0
4 Av! = -(4°-GAt9I)
Salil 0.75 0.25 -0.26
13 1)=| 025 0.75 0.25
oy) (ENP (is thee
mle
Note :A7A=I= A.A and hence A is correct. 923
golving A
Bye BY Inverse aM
sé
one ae aie
i ie! vector afd CH cong,
and out? € is to 4,
values of the ,,
Stem
Stix,
he correctness of the arte
va
¢
ty rules
a
uce " ample 35: Solve he
a y Bxt2y = 14
2) Sxt3y = 18
5 Com
golution: Given Equations fn 7m Th Bharathia: ite
: Altix fo, a
yec where the Coefficient, .
‘ n vector X = [x] BEER AY |
wns colum. \s) and the come
v
fi 3 3] AI 3
= ; = 8x3-2.9 — , %
A ( 3 x adj A (s rf
I 1f{3 na) 3} and
= — (adj A) = = ~
oe ees = 5 ( |
1 gives [|= 1/3 2) (QQ (a
- =A © ives () -2( 1d) _ (3)
; 3\3 3) lia) = \W
=2 and y=4
x
note: In ean), LHS. = 3.2104 = 14 = RHS
In eqn.(2), L.tLS. = 3x249.4 = 1g - RUS.
Hence, X = 2 and y = 4 are correct.
Example 36: Using matrix inversion method, solve the
loving systema. of equation: c.A, Foundation, M 96)
era Sa
x yatrix form ?
xeys
Solution : Given equations in ™
Te -1 $ x y
foo Wosce Yanda ltt
AX = G where A = uN z 4
eens |
regey = 1 cel
: Seen
A’ = [-3 1-1) aaj A= e i 3 *
ps el
JA[ = 2 x 24x 049 x (-2)= -2
-2 -4 eee 2
ee! -1 1].}o 058 -05 5
A EN (adj A)=— 1 3 1 CLS mene
x 4) 1 2 L 9)
0 05 —0.6))2)_)-1
X = A-!C gives |¥| = =
gives | 1 -0.6 -1.6][4 -6,
«x =9,y=s-landz=-6
Note:1.In eqn. (1),L.H.S.=2 x 9-(-1)+3x(-6) =1=RBS.
Tn eqn. (2), L. 9 + (-1) + (-6) =2=R.HS,
Ineqn. (3), LH.S.=9-(-1)+(-6) =4=RHS,
Hence, x = 9, y = -1 and z = -6 are correct.
2. For solving the equationg
2x + 10y + 22 = -28
4x
3x+9y + g=-28,
210 2 82 Sr 20
Lie | 2 aaa ea
56-1 36 12 -40 fe
3 i and B i a
ibe ; . c sf oy
Be B+A (ii) is Oe 2 that
ate GA = 2(B-2A) OE Des,
9B ¥. By
OK -ij1=B (viii) (Ba)- ie 570 GR}
he
* Ba
= By
Bile.’ 2 ter
5 7 ff andB. et
i: 1 - heel
fa: Ree Ot 2 ie ant
|
vite (4) @A2B) Gages) 45.
ie e(A-@B) = SALOB (wy (Ry MMi AR Tg
Wade =A Will) (AB 8) Bay. yy
Ten Tat
(oe = )—3 7 3 ee ‘|
ae 6 lb Reel
(ASB)eC = A+(B+0) Gi (Aye :
qi) A(B+C) =AB+AC (iy) (MBO AC eo
= a
pa tat
a © rr co Baek © (wi) (ABCY < oR s
i. et, 3
4 0 b e -2/ (19 13
Hee 2) 10 14 a
rind X and ¥ if X+¥ = 10 18 15\andX-Y=\0 1 ‘
Es f] 15 14 11 4\
Hint : X= (4)IK+Y)+O-Dh ¥= (4)icey-ow foe i
a9 If A la B
SA+SB+2X=
Hint: K=-$
10, ff A lo 2
(A+B)*=A*+2AB+E™
bee ace - [Re
act sila ot
the values of a and
sae
cen Ealio
-(:y.e-[ 4+
and b for which A+B
12. If A
of a
13. ae AB and BA when
23-5
ges 4 eal:
ad oe el be
)
14. (a) Te A = G |
(B.Com. (C.A) Bharathiar, Nz000)
and B=
fgnd X im the equation
z.C.S. Bharathiar, A209),
s oot ch. eee
(BBM. Bbarathiar, N
a2
1] ana A+B)" 2 Bee "
- (BBM. Bharathiar, Nag
and LC.W.A. Inter. J97
ZIP al, find the values 3
(.C.W.A. Inter. 599)
= BC.
10] z
ad iol
6
New
a
| 2
f 3| , find AB.
a
om
E 4) , find (AB) (A-B).
(B.Com, (C.A.) Bharathiar, N2000)
(b) Find BA.
Seat 14) and B=
peer |
_|6 2
ne. tf A*= [iF 24]
[3 -4
[s 4 ES
17, 1fA=/1 1/and B=
2 [3
Nites
» find AB.
B.Com. Bharataas Ag)
verify that (AB)"=BTAT
(B.Pharm. Madras, Ag%) 1)
927
Pes al [2 1|
Ee ‘ mee) Gi | 4
te @ 7 : 3| a a Ms A
poe a -1 |
a 2
i
Heed 3]
mele <2
Gia) Fi a
2 8
ex) 4
5 Pes Beer tC] = 9
aa = a
f B.Com. wea
: fr SSras, Wig)
axon’ whether the matrix Ps i aa 52)
i! Beles hs
~ © 3 2)~ sinew,
Com, (ay a Mewar
pind eee Be 1 sy Mon
i“ Avge T. La) @M4 chow 9
E . (B: nat
rind the imverse of Ba Madras, M38)
Y 4 2 i re
A ale i (MLA. M.K., A98) qi) a4 3|
fl ks aa L6 | MABdasan Nov
- iy Sa)
wit A =|7 a> find a matrix X such thet ax ;
a 4 me 18 a unit
matrix of order 2x2. (B.B.M. Bharathiar aes a
1 m4 and Mag)
Hint - Saat
: 5 4| fy 9]
. gglve the equation [ X= 2 Vise mal od ea
gP pais |) 3 | ere Xie a ae
(B.B.M. Bharathiar, 497)
5 al" (1 -2|
lea a
matrix.
Hint: Find [5
nlf A = a 1]
1
4 ES
a) and X =
show that A”-5A+7I, = 0
(B.Com. Bharathiar, NO4
,
1 2] o28
| prove that AzsatZL = 0
4| . prove
cy
o
2
o
@.gem. (GA) Bharathiar, A203) a
n that daa? -2dAtsel = 0 and hen,
show that 4
28, Given A = (LO.WA, Inter. Dos
(not otherwise) find Fie
-1
29.5 Pee [2 2 -1| satisfies the equation :
29.Show that the matrix’ # tee : ‘
AL a
3_ga*+9A-—4I = d hence deduce + jf
A®-6A°+9A—41 Oo an - ae pee rt (
| ie | Pp ;
30. If A =} Belpre agi A end hora ea
18 2 10) {
jer 3.
: is the null matrix of orc
where 0 is the x owl ee
I, 1 i
AS {3 2 Bc Opreve thee SG) )— (Acie s) ebay
{2 —1 3)
is a unit matrix of order 3
-3 -3|
o- 1| is itself.
rar
adjoint and (iv) determinant
where I,
. 4
32. Show that the adjoint of | 1
4
L
33, Find (i) minor (ii) cofactor @ii)
for the following matrix:
143
A= [2 2 | (B.B.M. Bharathiar, N95)
: [3 2 2]
34. Find the inverse of.
4° 0 2 5 6 4
@ A=/]2 10 2 dij) A=|7 4-3
3 9 1 2 18%
ae M. Bharathiar, Age and (B.Com. Bharathiar, N95)
Com. Bharathiar, 42000)
atrix A is give, Evia ca
, m
a f . 8 such that AB — > BA =
crc Gs. py
ce # Be "hors thins nee
B) : Cramer's rule Solve the following.
ir a
e 16; 3x+y=~]
oy (B.Com. Govt, College Kumbak.
: ~3konam, M2009
: —dy=38
r%) 3x ae : OLA. wok Non
+See 2 La Bidasan, non
: pomeemee ~2: xy
~ Sam
As
2
(B.Com Bharathiar,
pace ty)
x Trichy, N91 ang BBM. Bhay
ae so. SR ee :
B +52=11; 5x+2y—72~
23y
ASK;
+zZ=0; 2X+y—-Z=]- dy 4
oy
"4 athiar, Ng5)
le AX + By py ok
BR. M. Bha
arathi; > N97)
Spee) 25 x, ~x, 0 oy 173%,=0
3 MBA. 1eNou, Des)
=6, 3x
1-2x+3ye+z
7 - B=
3x0
i) 3
eed
C.C.W.A, Inter. D97;
16 ‘10x, — 2x, +X,=8, Sx, +
x, =0,
3
ot 2x, =7
a A. H-H.T.B.. eae N99)
6, 2+x=4 (1.CW.A. Foundation, D99)
poo] Fea ss
ist V= : a
; atrix method: i
—_ 7 (B.Com. Bharathiar, N95)
-x¥+ by mf.
3: - 3y = ae Bharath 97
\8.8.M.
(LO.W.A. Inter, Jag, Nie
(iii)8x + by =
ox + 26y + 362 = 16
B.C.S. Bharathiar, A200)
iv) x ty re =; 3
ee ae
(C.A. Foundation, M2000)
(vy) 2xn-y+3e= 7
44; axe Ty + 82-15 i
B.C.§. Bharathiar
(vi) x+3y 4 52= 10; 3:
(vii) dx, +2x,+6x,
(M.A. B’dasan, N37)
= 2
Set
(vill) x+y + 22=4; 2x—y+32=9; Oe
(B.A. Bidasan, Nog) le
to
ee ye t
(B.Com. Bharathiar, Mog)
=