Matrices
STPMSyllabusS&T
Matrices
Mathematics is not a book confined within a cover and bound between brazen c/asps, whose
contents it needs only
patience to ransack.
JAMES
JOSE1H SytysSfER
Learning outcomes: At the end of this topic, you should be able to
4.1 Matrices
b) use the condition for ihe equality of two matrices;
c) carry out matrix addition, matrix subtraction, scalar murtiplication, and
matrix multiplication for matrices with at most three rows and three
cclumns;
4.2 Matrices inverse
d) find the minors, cofactors, determinants, anO aOjoints
2x2 and 3 x3
rnatricesii
find ihe inverses of 2 x 2and 3 x 3 rron-singular matrice$
use the result, fcr non-singular matriffiiFnfillB)-r = gj:r 4-r
4.3 System of linear equaticns
s)
D
use inverse matrices for solving simultaneouJnnear equattonC
solve problems involving the use of a matrix equation.
(s 4 6\
l,
tt
Row, column and order of a matrix
(s
Io
-J
J
I
<-- row
-s) <- row 2
2'\
t
I
:
(f)
5
E
f
6
o
c
(e 'r B)
1
The order nf tiris matrix
......?.I..}........, and is atso
callerl a sguare matrlxLmatik
'"
segiempat sama.)
r0w
f
0rder' : zx)
.l
Airnpn;,on = 2x3
oo
Element s in matrix(Unsur-unsur matriks)
v
(olum fi
(:
l5
This matrix is said to be of order (peingkafl 2 x 3
1.
Matrix with m rows and n columns is of the order m
x n or the dimension of the matrix is m x n
Example 1
State the order of the following matrices
cotkMn I
+- Row i
+-Row2
4
/\
'l
I
I'
-6)
Ir.1<--wt
j-s l<- nrw:
I o J.- ^*;
fobJ 'ro ttrrl n
<_ Rowm
Example 2
The order of this matrix i" ..].X..1. ... and is also caled as
col um n matrix(m ati ks I aj u r.)
Given matrix
4 rolv.ttttt
,t.
(o ! 3 sFlruw
^
?'t't = 3
The orderofthis matrix i" .....1.I4.......,and is atso
knows as row matrix (Matiks bais)
Q2t= 5
=
the etements ror
,
[; ;J """
312 =
-Z)
d22 -
-("
c)
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n -
uL)
The nurnbers inside a matrix are called the
elements of the matrix.
For a matrix A, the element on ith-row and jrh column is denoted by a1
a)J
b)
o 2l
-1
By KKH
Matrices
STPMSyllabusS&T
(o"-b" o''-b"\
Ior,-b, or,-br, )
Irl Speclatmatrices:
3.
ldentitv Matrix. I is a square matrix with elements
on the leading diagonal equal to .l , and other
8.
Multiplication of matrices with a number.
elements equal to zero.
r(
r
I o,, or, )
( o\
I(0
^ t )lidentitymatrixof the order 2x2
(t o o\
I o t o I ioentrty marr ix of the order 3x3
tt
terrn.
_u(-2 5) _ f ro _2s)
Ia 3) [_zo _ts)
Example 3
(q 4)
i) ,=[; ;') ""1 t =lt -t)
Gue"A=[;
Example:
Le:tding
lko, kau )
Example:
(.0 o t)
Diaqonal l@!18 is a square matrix with all its
elemerrts equal to zero, except those on the
leading diagonal which has at least one non zero
o,, a,r\ _ ( ta,, ka,,\
Find,
a) 34-68 +9C
b) 7A-2(B-c)
diagonal has at
least one non
/.zero number.
L/
Multiplication of matrices
Svmmetric Matrix
9. The multiplication can or.;iy be performed on 2
The elements are symmetrical about the leading
diagonal.
Example:
matrices if the number of columns in the first
matrix is equal to the number of rows in the
second n':atiix.
The product of a matrix of the order m x p with a
matrix of the order p x n is a matrix of the order
mxn.
Zero Matrix( mafiks sriar) is a matrix with all its
elener rts eq':al tc zero; and is denoted by 0 .
equal
order 2 x
r---_-1
order3x?
3
I
Addition and subtraction of matrices.
7.
Addition or subtraction operaticn can only be
caried out orr matrices of the same order.
Addition of 2 matrices
( o,, br, * Q,rb^ t o,rbt,
lo^ b,, + arrbr, + ar-b,
( o,, a', ) ( b,, 6,, )
*
I o,, ou ) lb,, u, )=
( o,,+b,, o,, +b,r\
I, or, +
I
(b,, 4')
( o,, drz o,, ) u,, bul
=
( o, Qzz ar, ) I
{.4, b,, )
a, br, + arrbr, + a,r4, )
a, b,, + a.-br, + orrbr, )
I
order 2 x 2
Example 4
b, ar, + br, )
Find the product of
l)
(s o'rt(t
(a)l-.r3iir3l
Subtraction of 2 Matrices
( o,, au) - (-b,, )
',, =
I o,, ou.) (. 4, b,, )
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-2-
'[-2
6)
By KKH
Matrices
STPMSyllabusS&T
-
Example 7
(b) Given,
Find the determinant for each of the following matrices:
(,) A= (-2
3)
( s 6)
Egual matrices
Example 5
(ii\ B= (6 -3)
Find the talues for x, y, z, and w if
ysdrythat
'[:
;) = (], ,:-)
(z B )
.( a
x+v)
fz+w 3 )
leal=lAllBl
Determinant of a 3 x 3 rnatrix
10.
17. Fora3x3matrix,
[g drz o,, )
ttu
tw@
\4 o'2 )
I
Multiplication and addition of matrices is
associative
art
A(Bc) =(AB)C
A+(B+C)=(A+B)+C
11.
Consider the element ?21 , it the,row and column
through this element are crossed out, this leaves 4
elements which form a 2x2 determinant as below:
Multiplication of matrices is dlstributive over
addition or substration. *--
I
12.
Multiplication of matrices is not commutative
13.
A+(-A)=0
o,, a,,
I
lo" o"l
X1r-reY'=Ag"44e-A(B-C) =AB-aC
18. This determinant is known as the minor for
elemenl ar, , and can be denoted as Mzr.
AB*BA
Write down the minor for
A+0=A
-y.{ {
Mr=
M:z =
Example 6
ci""nA=[l
-',)r=(1f "*.=[; i)
Show that,
(i) Multiplication of matricee is associative.
(iD Multiplication of matrices is distributive over
addition
Example
'q
/ -. v i- i
Mzs =
(:l -:" - rr\
8
\t tt -t'v)
'*ri -l-v
1l --!Y
l\ I v-
- U
(s u -r')
civenA=i2 _8 s . Find the minor for each
(r -r -s)
I
Determinant of matrices
14.
elernent, and by using t6sss minors form a new matrix
where its elements is equal to its minor.
Determinant of a 2 x Z malrix
/'t
FindlheminorforB= I -
lnel=lAllBl
rJ.
(-2
o
'l ,n" determinanr
\c d)
For a square mat ix A = (
is defined as
'19. The minors, have associated sign, + or -,
Ia bl
lnl = | al,l =ad-bc
lc
16.
-r\|
a)
j+ - +l
lt +
l- +l-l
l+ -
The determinant of the producl AB is the same as
the product of the determinant of A and that of B
det (,AB) = dEt (A) x det (B)
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I
depending on the position of the element of which
they are a minor. These associated signs are
shown as below:
-3-
By KKH
-
Matrices
STPMSyllabusS&T
20. Hence, the sign for minor Mffi
+, and so_on. ln general, the sign for the element
ax is (-1)r*i
(r 2 3)
(b)
21. The minor of a particular element, together with its
associated sign is called the cofactol of that
element.
22" Consider the matrix
( o, drz o,,')
o=
Ion ezz or l
[o' Qtz ar, )
The cofactor of the elements are:
I
arr ls
o, a'l
lo, oul=
a,, iS
[find'the determinant by crossing the elements in
the first row and compare your answer for the
determinant by crossing the elements in the
second columnJ
(c)
c - (-?
25.
For a matrix, A , if its determinant
not has inverse rnairix. On the other hand, if rhe
lAf # 0, matrix A is a nongingular matrix and A has inverse nratrix.
Transpose of a Matrix and adjoint matrix
26.
Example 9
Find the cofactors for the following matrices:
:)
lo o 3)
23. Llenerally, cofac{or for the element q, is denoted
as A,, or ilil, cofacior for element a,, is denoted
as A,, cr ql? and so on.
Qrz 4,, )
24. For a 3x3 matrix, A - I o^ azz or, f , ,n"
['r,
Qtz
or, )
determinantlaf i"'
"
il I = e,r A,, - a,, A,, * e,, A,,
la." o,l _ o. lou orrl *
= o"
lo-r, artt, '- lr' arrl
o..
"'t !o"
o,,l
lo'" n"l
(a, ar, - a, e.rr) ao (quett-dt, urr) + c:,r(a,arr-a,arr)
= ar,
For a matrix, A, if all its elements are reflect on the
leading diagonal, then the new matrix formed is
called the transpose of matrix A and is denoted
ov AT
27.
(t -r 2\
b) v=lo 5 6l
( u,
lAl = O, ,utrl<
.leterm:nant
";,:,1= - (o, or, - a, arr)
"= [;
3')
(s 6)
A is said te be sinqular matnx and matrir A does
ct22 Q$ - Q32 A23
1:,:
n,,
al is -llo,, I - -(o,, or, - a, a,r)
o, o,rl
a)
e=l+ o 6l
lt(7 0 e)
The transpose of a matrix A is the matrix whose
iows are the columns and whose columns are the
rows of A, that is, the rows and columns are
interchanged.
p=
thd
(s 2 3\
Pr= f' 4 -tftransposeore
fo 4 s)
Example 11
Find tl^e transpose matrices for:
(t 2 r)
a) ttl: -l 4l
(0 -3 2,1
(s 5 ol
b) lo trl
(-2 4 6)
Example 12
Example 10
Find the determinant for rhe following matrices;
(z ' -r)
(a) o=lo -r--tl
[3 2 4)
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|
(t 4 6)
6;,r"nx=f o 2 3l
l.-' z -s)
a) lf cofactorofX, oi; = (-1)i+j M4, where Mx isthe
minor for element xij of matrix X. Find ali the
cofactors for X.
By KKH
STPMSyllabusS&T
-
b) Form a new matrix where its elements areThe
conesponding cofactors
Ithis matix is called matrix of cofactors forXl
I
= A-1 A = l
28. lf every element in the matrix transpose, Ar , is
replaced by its corresponding cofactois, the new
matrix formed is called adjoint matrix for A and is
denoted by adj A.
Example:
4)
_t
I
M= I 3
lo
'
[+
3)
J
o)
1
ll
5
3)
=
(: l) ,"""
I
I
A does not exist and A is called a singular
matrix.
Example 14
I
-(-2)
^
32. lf the determinant of A is equal to zero, inverse of
( -o-s -(3 - 4)
5+8 \
adj M =
-6
-(-r0- r2)
l<n-ul
Y
31. rrA is a non-sinsurar;"natrix,
I e'= r (.-(a -b)
,"-r, c a )
I
r_r
and
(l
lnverse of 2 x2 ft4atrices
the inveise of matrix A is defined as:
( -'t
Mt =i
inverse matrix for A and is denoted as A-1
Similarly, matrix A ic also an inverse marrix for B
and is denoted by B'1 .
Thus,
Adjoint Matrix( adi A)
(-z
where I is an identity matrix, lhen B is called the
4-3 )
,o=(u d)
?l
(-tr I t3\
=l-s -6 22l|
ic
ql,ru"nrhatAB=r,nnd
u=(,
""0
s)'---"
lr
matrix B in lerms of a, b, c and d.
It , tl
Example 15
2-a. The adjoint of M can also be determined by the
s)
[2
rno
t.4 6)
6)
e=(( t _2.4)'
-2
following steps:
i) Find the matrix of cofactors for M
civen .q=
Mairix of cofactors =
Delermine which is a non-singular matrix. Find the
inverse matrix for the non-singular matrix.
(ii) Transpose this matrix to get the adjoint
Example 16
matrix.
(z 4 :)
oirenA=l r
adjM =
[-r -v -a)
Example 13
Find AB and hence deduce A-1.
Find the adjoint matrices for::
(s -2\
(a) A= i
z -rf ano
t-2 -3 t)
(t r t)
B=l_r _8 _sl.
Example 17
'.4 -2)
I
l)rnae=[lo)
(o 2) Find matrix X if
13 2)
(z
(s 2 3'l
(b)o=ll o -'
[o -3 s)
Given A = |
AXA-I = B.
l
I
Hence, find Q adj Q.
Example f I
/'> -r\
^' I and I and O are identity matrix and
[-5 3)
Given A = | '-
lnverse Matrices
null matrices respectively and their dimension is 2 x2 .
Find the values for m and n such that
30. lf A and B are 2 square matrices such that
A2+mA+nl=0.
AB=BA=l
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By KKH
STPMSyllabusS&T
Hence. deduce A-1 and A3 .
[matrix A is refers as the zero
or the rcot for the above quadntic equationl
lf we use A tc represen
4 c)
ts natrix(:, b,
(.*
Example 19
/-x
lf A, P and D are 2 x 2 non-singular matrices with Ap =
PD, prove that
A2 = pD2p-l ,and
A3 = pD3p'1.
\
to represents matrix I y I .no B to represents
l,)
(d,\
4 l; men, ih-e ahove nratrix equation can
Ir, ]
matrix |
lnverce of a 3 x 3 Matrix
beexpressedas: AX=B
33. lf M is a 3 x 3. non-singuiar metrix with the
36. To deter.mine the values for x, y and z , we just
rraed t{J the ;'natrix equation as below:
AX=B
A{Ax=A-1 B
X =A-1 B.
Example 23
(z
I
lf e= I -2 -'rl
""0
[, 2 o)
Example 20
I
Determine the following inverse matrices:
(r -2 t'l
(r I t)
(z
"=l; :,:J' '=li ) :,)
B=lt3
;l
l.*
s)
Frnd AB.
Hence, solve the following equations.
2x+y+z =1
Example 2l
x-2y -32 = 1
2y+42+3x=5
If A and B are non-singular matrices, show that, (AB)'1
=BiAi
'j-.,.. ::.-: I
.;^'?.iil
Special cases
:
Example22
Case 1
Using the inverse matrices for C and D in Example 18,
Consider the following linear equalions:
deduce (CDf ' .
3x+5y=4
6x+10y=8
The matrix equation is
(t ') [') = l,n)
System of Liirear Equation
[o rc) \y) [aj
AX=B
34. Consider a set of equations with 3 unknown x, y, z:
alx+b1l+CrZ=dt
The dcterminant f Af = 3(1C) - 5(6) = g
a2x+b2f+CzZ=dz
Hence, matrix A is a singular matrix and the inverse
matrix is not defineC.
a3x+Lr3Y+CaZ=de
lf we pre-multiply the matrix equation Oy (10 -s') , we
This set of equations can be expressed in matrix
form as below:
l-o t )
get
(o, 4 ",) f-) /,0,')
Io, b, ",lly l=lazlmatrixequation
|.* bj ,,) l,) [.orj
[]: ilt: ,i) (;) =
[tj
'
(o ot (') - (o)
oJ
i
lo
[.yJ
[.oJ
There is no unique solution. The solution set is
{ (x,y) : 2x+ 3y = 4l
Geometrically, lhey represent the same straight line.
Determi4ant of matrix
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l,*
"
;, ;,)
)
-6-
By KKH
-
Matrices
STPMSyllabusS&T
Case 2
Consider the following linear equations:
x-3Y = 2
Exercise 1
3x'9Y=5
The matrix equation is
(t tl r,) = r'')
(: -g) ly) [s,i
AX=B
The determinant l,t!= 1(-9)-(-3)(3) = A
Hence, matrix A. is a singular matrix and the inverse
matrix is not defined.
if we pre-muttipty the matrix equatio n oV (-
n ,) , n""
ll
t(")&(,,) (s
[-r t)
9et
[-] i)tl i) (;) =(-; i) t;)
z. o,"""o =
I
I
a) AB
b) BA.
r
(; ;) (;)=[,')
[i
l
_tj] ""..
=
[] ; ,o)
o"o
c) ls multiplication of malrices commutative?
(-r -4 -'r)
r -2 -o i ro= (lf _1,) ,,"0 *o,
There are no finite solutions for x and y. This can
be seen geometrically as 2 parallel lines.
tAB= |
[-: -6 -20) '
Example 24
3. n*= (t. 2)
^2) uno
(t
tt -z r\
[4
(a) 114=120 t2 6l,nnol't
(o o
I,onothevarues
'-"^= (5 _3)'
for h and k such that HK = KH.
lt
\le te 0)
[h:714; k= -512]
(b) To cerebrate Hari Raya Aidil Fitri, Amir wishes to
sent 38 new year cards to his fiiends.
From a book shop, Amir selecled 3 different types of
cards, which are pricbd differenfly.
The price for the second type of card is the me:n price
:
for the first and third
He has to pay RM18.1C for buying 20 pieces of the fir st
type, 12 pieces of the second type and 6 pieces of the
third type.
lf he has selected '19 pieces of the first gpe and also the
second type respectively, then he has to pay RMl7.10
.ry.'
civen s =
os)
[f
]J. ,,"0 s-1. rrr = (-ts
'
ir )
[-ro 40) "nc
3
type.
l'
By assuming that x, y and z are the prices for the first,
second and third type of card respectively, obtain a
system of linear equations which represent this
information.
Write this equations set in matrix form, and hence find
the price for each type of cards.
[35, 55, 751
I
(
l),"to*,n"t,T=scsi. Hence, show
r
'
ihai T3 = Sc3s-1.
Find M2", *= (1 ]l . I i, an identity matrix,
[2 3) 't
"=
c'ider 2x2 ano 0 is a null matrix of the order2x2.
Find the values for h and k such that M2 + hM + kl
=Q.
Hence, find M-1.
w'= (:: ll); n=-t, x=21
I
r
lsi
I
r
I
Show that,' A = ( '. '^ J ,, -.ro for the porynomiat
\,4 _3)
sifar g(x) = f
+ 2x- 11. Hence find A-1
rn''=-L[3
']'
'
ll(.i -t)'
o)
,rA =
[-', i ;l showthat ^'= f ; i ul
l: 2 t)
l.-s -2 t)
(r 4 -2)
(t -4' la\
l1s=lo r I j,snowthatBl=10 -,I
I
lo o t)
Hence or otherwise, find .(AB)'1.
[o
o
')
;,
/
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-7 -
By KKH
\Matrices
STPMSyllabusS&T
,/ (-rt -32 )
i Ilt6 7 -3ll
\ l-s -2 t)
By,.ysing r.aE.y to represent tne pront oOtaineO
Uy
selling each'prince' and .Dunlop, iackel
respectively, write a system of linear equations,
which represent this information.
Rewrite this equation in the form of matrix, and
obtains the cost for each type of rackets.
14
8.
/_)
lf the diagonal-rnatrix A = |
l. o
l),t".A2 and
43 . Hence deduce An.
A'=
; I (;) = (:;:);x=1g,v=t, RM81; RMssl
o"=(," oj,
(: | o' = (-i :,)
,
Show that, the inverse matrix for
(r r-t) (-s--s 5)
Iz 3 2li'll
.l'
| tOl s 6 -4|
l_
y GivenU=f]l ,=(-r'J ,nor=f7
-r)
'
(r/'
/
lz)-"" [-z a)'
0
(5 1 t)
Find TU and TV. Hence find '1. U and Tn V in
term of U and V where n is an positive
integer.
lakca, mi rebus, and nasi berini. The price for a
plate of nasi berani is equal to the sum oi the price
for a plate of raksa and mi rebus. A group of
workers paid RM'|6 lo buy 2 plates of laksa, 3
plates of mi rebus, and 2 plates of nasi berani.
Another group of workers, paid RM19 for 5 plates of
laksa, 4 plates of mi rebus, and l,plate nasi berani.
By using x, y, and z each representing the price for
a plate of laksa, mi rebus, and nasi bLraniy
respectively, obtain a system of linear equation to
represent this information. Rewrite this equation in
the matrix form, and get the prices for each plate of
lal<sa, mi rebus, and nasi berani.
and y. Hence, showthat
f*) = 3n-l ( z"o*+ y)+3n f*-y) )
(v/
[zn1zx + y)-3n ex-2i)
t(i) ru =
/o)
[uJ'
w=
Tnv=9nv (ii)o =
/- q\
=6n u;
| ti );r"u
,l: i ;'']1, l=[,?'], ,=r so, y = 2 oo;z= 3 5ol
!{z*+y);g= }fv-xl.t
1P. Using matrices, solve the following simultanecus
'-'/,/
\5
equation:
al 2x-3y=13 ; 5x+8y-34
b) 6P+5q+9=0; 3p-7q +33=9
c) 3r+6s='lC;9r'-3s=2
(t I l)
(t
4 o\
A=lt -r -rf ano a=l zs -n -31,
r.t 8 t2)
(_lz 7 3)
find AB and hence d","*tn" Ot.,,. ,', _ .
1,1'. By us;ng matrix equation, solve ihe iollowing
simultaneous equction, give your answer in terms
of k. State the range of values for k fr.rr these
anslvers to be valid. Find, the solution wh;n k = 4.
-/
3i<x-3Y=9
A is a2x2 matrixendA, - A * 2t = O
1? Given,
where I and O is matrix identity and null matrix
J", respectively. The order of I ani 0 is 2 x 2. Deduce
houses built and the associated
Types of
No of
Profit from
house
each house
Low cost
X
RM1000
Terrace
RM8000
Y
Semi-D
z
RM12000
The developer nas
million for this project.
Based on this information, form a svstem of linear
equations, and rewnte these equations in the
matrix form.
Solve this matrix equation and determine the
number of houses for each type that chould build
by the developer.
that, A' + 34. + 4l- 0.
A man spgnds Rlv184 tc buy 100 paper folders,
which consists of 2 Cifferent types. The prices for
each type ofpaper fclder are 70sen and g0 sen
respectively. lf I and y represent the quantity for
each type of paoer folde;s bought by the man,
write a pair of simultaneous equation to represent
this information and then form a rn:trix equatio;i
and solve for the values of x and y.
(t r\/-r\
1/. A sport shop sells two types of racket, the 'prince,
.//
and the'Dunlop'. The selling price for each type of
racket is RMv9 and RM79 respectively. lf the shop
sells 5'Prince' rackets and g ,Dunlop' rackets, it
*if,T"!9 a priofit of RM258. whereas, if the shop
,Dunlop'
Tsell 8 'Prince' rackets and 5
rackets, the
profit rnake is RM249.
@All rights reserved
oot@.9
/roo\
It n)l)= IrooJ;x=so, v=7ol
;
ln a housing project, a developer plan to build 300
low and medium cost houses. For medium cost
houses, there are two types_ffpt$Jg6l@?nd
semi-detached house. For eveg 2 m-eOium cost
houses build, the dEvelopei inuit OuilO a low cosl
house. The following lable show the number of
(k+1)x+ 4y=35
t
t r )lz) t1s)
16. tf
(a)x=206/3'i, y=31J1: (b) p=4, q=3; r=213. s4l3l
. ,'
i)
A fooo stores sell 3 different types of food, that is,
(ii) tf i/x\
+
_. I = o U F V. Find o and B in term of x
\v/
T,,
\_7 t
,
( q 4 o\
tl n o l;n.'=,Lf zs -n -rl
(o 0 t2)
\_t7 7 3)
(t t t)/x\ /:oo\
-r -, ll , l=l o l; x=100, y=1so; z=5ol
fr I rz)\:) [re00J
(rz o o)
,
l,
|
-
STPMSyllabusS&T
A man pays RM4.20 for 3 cups of tea, and S cupl
The total price for 6 sweets with fruit favour, g
sweets with coffee favour, and 6 sweets with
coclate favour is RM3.00.
Using x, y, and z to represent the prices for a
sweetwith fruit, or coffee, or coclate favour
repectively, form a matrix equation for this given
information. Hence solve the matrix equation to
determine the prices for each type of sweet.
of coffee in a coffee shop. On the following day,
he pays RM2.60 for 2 cups of tea and S cups 6t
cof.ree, in the same shop. Obtain, a sel of matrices
equation to represents this infornration. Hence,
find the price for a cup of tea and a cup of coffee.
//
=(w; RMo 4o; RMo 60l
;)(;l
{._(:
(s o r)/;r\ /zoo\
ir rcl A restaurant,
I
! /
l, i .
\-'
offers 3 gpes of roods, that are roti
canai, sate and mi goreng. price for each plate of
food are fixed, and the price for 3 plates of roti
ca1,a1g
lnegame as a ptate of mi goreng. A fanrty
paid RM38 for 1 plate of roti canai, 3 plaies of
plates mi goreng. Another family, paid
19te, and 2
]RM52 for 5 plates sate and 2 plates cf rni goreng.
By using, x. y and z to represents the pnce for a
piate of roti canar, sate and mi goreng respectively.
Obtains, a set of ;'natrix equation to represent this
problem.
Solve this matrix equation, and find the price for a
plate of roti canai, sate and mi goreng.
'[; l(r=[if ''=,,
19..
(-+
z= 2Osenl
Shows ihat, the product of nrairices
-8 s)
(t r -rj[-z+
q.
ll0 .s 3 il 15 -13
1,.,r r o,ll.r t u)
|
can be expressed in ihe form kl, whe;e k is a
constant and I is a 3x3 identity matrix State the
value for
ft=16J
The Parents and Teachers Association(pfny bi tna
Wahab Secondary School, plans to help the poor
students in their school. The pTA's committee has
decided to donate not more than Rtvl1000 a year to
help the poor students to pay for their pMR, SpM,
and STPM examination fees. The amount given
out to individual student is differ for students at
different level. The total amount given to a pMR
and a SPM student is the same as the amounl
given to a STPM student. Last year, the total
amount given out to 10 PMR students, 5 SpM
students, and 3 STPM students is RM925. This
year, the tolal amount given out to 15 pMR
students and 8 SPM sludents is RM975.
By using x, y, and z to represent the amount of
money given to a PMR, a SpM and a STpM
student respectively; form a system of malrix
equation which represent this information. Hence,
solve this matrix equation to determine the amount
of money given to a PMR, a SpM and a STpM
student
k.
v=8, z= o]
(t -2 r)
(o 4 -4\
[., , t)
[-, -; 6)
6a=lz z rfanoa=l r -z r l,nnoee
and hence, find A-1.
A fruits hawker sell oranges, apoles and rambutan
in his store. The selling price for an apple is the
rnean prices tor an omnge ano ramputan.
His first customer, paid Rf,115.50 for l0 oranges,
10 apples and 5 rambutan.
lJis second customer, paid Rt i38.00 for 30
oranges, 20 apples, and 10 rambutan. Assume
thai x, y and zin sen represent the price for an
lrange, an acple and a rambutan respectively,
build a syste;n of linear equation, which represents
this information. Rewite the eguation in the form of
rnatrix and hence solve for the price for an orange,
an apple and a rambutan.
0 0\
(o -t t\
(r
tl ro
rlo 4 of;a-'=f-* + i+l:x-zy+z-0.
\0
rlo , -llrl-l o f;x=rosen;y=15sen;
[3 4 3)l..-/ [r5o.l
-r)fj) f o)
t' l ll tl=lezs
;x=RM25;y=RM7s;
8
o -4) (.+ 2 -+)
2x+2y + z = 3 1 0 : 3x+2y+z=380
(t
lf A=
I ii]t'l [il]
-/
l.'
'l'
-2 -a)
shows that AB = 101, with I stands for a 3 x 3,
identity matrix.
To celebrate Hari Raya, Mr. Hamid plan to buy 3
different types of sweets, with fruit, coffee or
coclate favour. He has to pay RM2.00, for buying 5
sweets with fruit favour, 6 sweets with coffee
favour and 3 sweets with coclate favour. The price
for 4 sweets with fruit favour and 4 sweets with
coffee favour is the same as the price for b sweets
with coclate favour.
OAll rights reseryed
: j.] ""'= f-: -i1 -j] '"'^'
Hance, deiermine A-1.
The Wahab Secondary School plans to encourage
their studenis ro :ultivate a gooo reading habit. To
achieve that, the school orders 3 different types of
magazine monthly for its school horary. The types
of magazines order are: The Current Affairs
Maga:ine, The Economy Magazine, anj The
Scrences Magazine. The total cost for a copy of
The Economy Magazine and a copy ofThe
Sciences Magazine are equal to 4 times the cost
for a copy of The Current Affairs Magazine. The
total cost for 3 copies of The Cunent Affairs
Magazine,2 copies of The Economy Magazine,
and a copy of The Sciences Magazine is RM43.
Whereas, total cost for 2 copies of The Current
Affairs Magazine, 4 copies of The Economy
(s o r\
(n _6 qz\
(a)lfA=la a -slanoB-l-zt6
4s
.r'
r
r
r
(3 4 3)
lt
I'
70sen,60 sen 50 senl
I
,/
f
0,1[:) lets)
[15
z=R Ml001
-9-
By KKH
F
Matrices
STPMSyllabusS&T
o@
lul1s.qltu, and 3 copies
is RM78.
By using x, y, and z to represent the prices for
a
copy of Tfe Curent Affairs Magazine, The
Economy Magazine and The S6iences Magazine "
respectively; form a set of matrix equation io
reprcsent this information.
Hence, solve this matrix equation to find the values
for the price of each type oi magazine.
(t o o)
?4. rf p is a matrix such thar p = (;
product ppr,
elements such that,r, =
-z A ul,
=
r
(t -r 3\
D A=12 o ol
(e -2 22)
(-t o 4')
ii) B=f 6 s -21.
lr 4 -t)
(z -r +\
iii)c=13 2 il
lr -z ,)
1 jl\ i" ,"i.r to uu
fi
^,
* rn"
[o o .,{)
upper triangular matrix or a eselon matrix. Find an
eselon matrix R with positive diagonal elements
(o _u 2)
_
such that RrR = I -6 t0 -l L wnere Rr is the
-[, -3 rc)'
R.
,[3 l' ,''
, J'
3le
rhgmatnx t =
[: :), ^tt,is2nidentity
matrix.Showthat, ila'.d * 0,then *=
*t
M2 + (ad
- bc) tl.
show atso tn"t , itru'=
-4\
,1.
(l 9, rq:., =
1a+d)2 ,
!n?t,pl" = 1a+d)z -2(ad-bc) and ps - qr = (ad -
-4 -4 2)
(t t6
r\
I
"=[i ')I
l.o
l-' 2 5)
-23:adjt=
A 3x3 matrix
0\
The matrices P and Q are such that pe ep.
=
Show that (P + Q)3 = p3 +
* epd * 03.
lz s t)
(z 3 t\
ur e=li i il , ':
16; adj n=f -zo 4
bc)'. Hence, find 4 distinct nratrices M such tirat M2
-, -3 -ttl
-' )
'\-5 4
(to
r
-z)
o; adjc =l-sz -2 t4 ii r,o
(iv)
l-'u -l :)
(r -2 o)
3;adjD=lr ' 0l
\_7 _r 3)
( t _r3
-60; adj e=
!-zs _7I
r
[13
Revision Exercise-Past years euestions
b
Find al the matrices for A such that A =
:l
(; ,r)
and A(l - A) = 0, where I is a 2x2 identity matrix
and O is a 2x2 nun
jt SSSS1.+1
matrix.
-e)
'
@All rights reserved
thal
f,. .f ) n^" positive diagoiral elements that
transpose or
(r 2 o\
iv) o=f -1 r ol
0
^^overifo
(a o\
,(:, r"i,)),'=[; l),
For the following matrices, fintJ
tb) The determinant of the matrices,
b) The adjoint matrices, and
c) The inverse matrices if it existed.
t6
[,n,
QQr = Prp. lf R = erp-l, verifo that RRr is an
identity matrix.
3 2 I ll ;, l=l r: l; RMs; RM8, RM i2l
I
1.2 o t)\'i [zsl
(t
,n1)
p'p = (25 20) Find also,
the malrix e where e
l20 2s)
[o o 7) 'lr _i8 n)
(t -r -')f,) f o )
23.
pr
where
is the iranspose of e.
Hence find the matrix of p with positive diagonal
(z _t r\
Jae=f o z ofa'=1f
:) evatuate the
- 10-
(; ;) [; i) ti
i [:; ,jj,
By KKH
:
STPMSyllabusS&T
n
-/
o=(u d)
l),o' = [* Y);rnothevaruesror
\c
\z
w)
x,y,zw in term of a, b, cd.
A tailor sells shirt anC trouser.s. He charge Mr
Hassan RM300 for 2 shirt and 2 trousers; and Mrs
Handi paid RM675 for a shirt and 6 trousers. lf x
and y stand for the pnces for a shirt and a trouser
respectively; cbtains a system oi linear equations
in marrix form to represent this information. Solve
this matrix equation to determine the price for a
shirt and a lrousers.
_d
lX=
'
-b
ad-bc
a- .(l
"a-u'
[r
,t=--------:-,Z=
-ad-bc'
[1e9ss1.1i]
-c
ad-bc
-,w=
:)fi)= t:;:) shirt = Rt"r45; t'louser
= RM1CSl
[19935i.11]
3.rt Mairices for A and B is Cefined as
(t o o.) /"s _s\ 01 7
rAB=sl 0 I ol,"'l-r o 5l
[o o t) \1 , ro -s)
(+ , ,)f-) [:ooo'1 \.*
"
lz 2 2ll v l=l zooo l;500rs,250ks,250 ksl
(z
(x v\
,/.// A=l 3)I ' B=l''l
(-r t)
(r -t)
[+ 4 2)lz) [:sooJ
where x and y are real nurnbers. Find the diagonal
matrfx Dr.such that ADA-I = B.
- lr ,
A company, with branches in these 3 towns was
given a contract, worth RM5000 in each town, to
supply x kg of lada merah, y kg of kacang panjang,
and z kg of mentimun in each town to the locai
retail sellers. The commodities are acquired from
the town itse-lf.Jheprofi{obtained by he branches
from Kualq/f ere'irgganu, fuala Lumy'urj and Johor
Bahru is ryM20p0jRM3Q00, and RIr[Ub0
respectivdtpdbtahrEa/riratrix eq,'atidn in terms of
x, y and z to represent this information.'
Hence, find the quantity-gf lada merah, kacan,o
panjang, and mentinrurr supplied from each tcwn.
(-z o'r
1'=o;g;D= l.
e ))lttss+st.+l
(r o o)
| (a o o)
4. ,f,tlalrices P and e is defined as
/
,/ p=l-r
-2 0l ; o=iu
-5 ol
'
|[-l -3
t.t
(c -3 -r)
(t -r r)
7. na=12 0 I l,showrhatA3=A. Hence,find
[o 2 -t)
A4o.
[1996s1.4]
(-t r -r)
reoo=12
o,lr
l+ -2 3)
i) lf P = mQ + nl, where a, b, c, m, and n are real
number and I is 3xii rnatrix, iind a, b, c, m, and n.
ii) Show that, nratrices P and e obey the Law of
5,
u=('
d!,'
.
commutaiive.
iii) Find the reai numr"rer s and t such that f = sp
+ t l, where I is 3xi identitv matrix.
Decluce that, Pa = - 5P * il.
[199451.111
l(i) a:2; b= -'t: c = -1; m = 1,
n=3; (ii) s=_1, t=2J
lf rnatrix iyl is defined as
(6
J
2
)
^-t)
Find the set of values for M such that inverse of M
is defined
[{m: m eR, m *-3,m t +})
[199551.4]
6.-,,'. Matrices for A and B are given as
./
,/
[s -5 o) (z' ,')
e=l_5 | 5ls=ll I ll
loro s) [z2t)
Find AB and B-1.
Ti,e following table shows the whole sale price, in
RM per kg, for s ty,pes ui commodities, in 3 major
towns. These commorlities are: Lada merah,
kacang panjang, and mentimun.
Commodities
Towns
Kuala
Terenooanu
Kuala
Lumour
Johor Bahru
@All rights reserved
MatrixA is given
nl= 0, where m and n are real numbers, I is a 2p.
identity matf$, and 0 is ?&.zero matrix, find the
values for m and n.
[199751.4]
[m=3, n=-2J
Matrices A and B are given as
(o o ,)
(r, -s -'r)
lr 3 o)
[-o t4 _d)
o=lr t rl.B=l-r I ::l
Find AB and A-1.
A company manufacture 3 types of lnstant Coffee
under the lrrand name: Kopi Jerai, Kopi Ledang,
and l(opi lvlulu. The ingredients for all these lnstant
Cotfee are coffee powder, sugar and cream which
are mired vrith different percentage (according to
their rrrass) as shcwn in the table below.
Percentage c:mpositien
Brand
Coffee
oowder & sueiar
Kooi lerai4
Lada
Merah
Kacang
oaniano
Mentimun
4
2
2
2
2
2
4
4
2
-2J. not- mA-
""a = l,1
U 0)
Kopi
,1
Ledano '
cream
60
30
10
40
30
30
Kopi Mulur,
30
70
0
The company plans to market a new type of instant
coffee by mixing KopiJerai, Kopi Ledang, and Kopi
Mulu under the brand name Jelemu. The mass for
each pack of Jelemu is 50 g and contains 44%
coffee powder,38% sugar, and 18% cream
By KKH
-
Mairices
STPMSyllabusS&T
powder. lf every pack of Jelemu contains x g of
Kopi Jerai, y g of Kopi Ledang, and z g of Kopi
Mulu, shows that,
remainder have to provide services within Melaka
itself.
i) On a certain day, if the number of taxis in Kuala
Lumpur, lpoh and Melaka are represented by Ko,
l',) [,J
represent the number of taxis in Kuala Lumpur,
lpoh, and Melaka on the following day. By using
this information, form a matrix equation which
relates Kr, L, and Mr with Ko, lo, and Mo.
ii) lf on one Monday, the number of taiis in Kuala
Lumpur, lpoh and Melaka are 100, 20, and 40
respectively, find the nuirber of taxis in Kuala
Lumpur, lpoh, and Melaka on Thursday, in the
same week.
iiD lf K", ln, and Mn represent he number of taxis i;r
Kuala Lumpur, tpoh and Melaka respectively after
;r days, find a matrix equation',vhich relates G, ln,
and Mn, with 1$, lo, and Mc.
[199851.111
'
(r'\ /zzo\
alrl=l'rol.
lo, and Mo respectively , and Kr, lr, and Mr,
Hence, find the mass ofkopi Jerai, Kopi Ledang,
and Kopi Mulu in each pack of 50 g Kopi Jelemu.
[199751.11]
(to o
1ee=lo ao
[,r
o
2l -9
0\
o l;a-'
80)
[159, 25s, 1OgJ
1p' (i)i"latrix A is defined .s
,/
(r r :\
a=lo
tt 2 zl
[-r t 3)
.
'
li' li,i lY'.l, riee8s1 11]
lr,.)=[,];l 0.05
0.8s)
\M,) t0.07
[-l
I
ilt, t;]
(z , ,l
lvl=l-l 0 4l
[., -r t)
ShowS that
wt3-3M2+8M-24I=0.
deduce M'1 .
(b) Matrices A and B are given as
0.
."tirty, A2 - A
( I .31
t)
(-s r 7\
[:
lz
Find AB. Hence, solve the follovring
simultaneous equations
-5x+y+72=8
x+7y-52=-16
7x-5y+z=14
(q -4 12)
-
3l = 0. Hence, witlrout fnding, A3 cr A' , shows
..'--'-_\at, A" =7A,+ 12t
[199851.4]
,/
\'
12. Eyery day,160 taxis which belongc tc a company,
/
,-ttre allocated arcording to the need, to providc
\\--l
either inter-cities nr intra-city (within a same city)
-rl
") 1M''= -l24t 5
(r
,/
I
5
,.=
services. For taxis in Kuala Lumpur, 3% of them
have to travel to lpoh,7% have to travel to
Melaka, and the remainder have to provide
services within the Kuala Lumpur City. For the
taxis in lpo[--S% have to travel to Kuala Lumpur,
5o/o go.lo Melaka, and the remainder travel within
lpoh city itself. For taxis from Melaka, 10o/o have
to travel to Kuala Lumpur, 5% go to lpoh, and the
@All rights reserved
(t 2 i\
t=l' , -tl
lr r'l
| 2)
-i t)
RM5, RM2; RMll
11. Shows that matrix A = I
(; ;)
(a) Matrix M is given as
r
2
\Mo)
l)' ' = (: ,'), "'o " =
3. lf A=
(q
(q -r -2\
-2)
tt-tt -r
ttDl-z t -rl,A.'=*l-2 t -rl
t,
l)
-1 )l
\- -)- )l-/
ri'llo
j t(t
" [.il] ;,1 :::;il;
;KL= 87,lpoh= 28, Melaka= 45;
Find A2 - 64 + 111, where I is a 3x3 identity matrix.
Shcws that, A(A2 - 6A + 't 1 l) = 61, and deduce
A-,.
(ii) A factory produce 3 types of ball bearlng and
market them under the brand name Kuda ball
bearing, Botak ball bearing, and Parang ball
bearing. The profit derived from selling 1 kg of
Kuda ball bearing, 1 kg of BoJaleball bearing and 2
kg of Parang ball bearing id-RMg" Profit obtained
from selling 1 kg of Botak bElf-bea:'ing and 1 kg of
Parang ball bearingis RM3. The profit gained from
selling 1 kg of Botalitalt bearing and 3 kg of
Parang ball bearing is the same as the profit
gained from selling 1 kg Kuda ballbearing.
By using, x, y and z stand for tre proft gained frem
selling 1 kg of Kuda ball beadng, 1 kg of Botak ball
bearing, and I kg of Parang ballbearing; obtains a
system of matrix equation which represents this
information.
Hence, solve for the profit derived from selling 1 kg
of Kuda ball bearing , 1 kg of Botak ball bearing ,
and 1 kg of Parang ball bearing .
(t
( o.s o.o5 o.l )f & ) t',(, ') rbw
'w
j=l.^l
(rs o o)
-sl;tollo 18 ol
3)
[o o r8]
25.
3 [199951.11]
Matrix P is defined as
P--
'2 r)tl
t t)
12-
By KKH
STPMSyllabusS&T
Find matrix R such that R = P' - 4P- l, where I is
a 3x3 identity matrix . Showa that PR + 4l = 0 [200051
(r r -3)
1n=l r'-: r I
/
t
[-,
(3
[2
4
lo : 1l['] [f]
''
/ l2a-t
,, (1t-t a
{. b
of shirts )
4
3
2
2
2
4
4
3
190
295
2
250
(-z z -i)
[0 0 t)
[2 -6 5)
=
17. tfA=!
1=te,y=ae 7=rgl
(l 2\
2'r
landE=l 3)l,findmatrixC
(-3 4)
u
Z/ sucs'rthat A = BcB-l.
./ ,^ _ (t o'\,
,t"-lo 2)'
'/
,*g
(,) [ zso )
Matrices A and B is given as:
(-l 2 2\
[+ o -t)
[-+ 8 t)
A=1" ,l,t=lt
lru
-J -31
Find AB an.r deCuce A-1.
ln conjunclicn with the in coming SEA XXI game,
the Wawasan Company, sells 3 typcs souvenirs,
that are: kev chains, Calculators, and Pens. The
ccmparry orders these items in two different packs.
The cost of a pack which contains three key chains
and two calculators is RM45; whereas, the cost for
a pack which cont:ins onc liey chain, one
calculator, and a perr is RM40. Cost for a pen is
four times the cost of a key chain. lf the cost of a
key chain, a calculator, and a pen are RM x, RM y'
and RM z respectively; obtains a matrix equation to
represents this information. Find the cost for each
individual souvenir supplied to the company.
The selling price for each pack of souvenir is fixed
at RM80. lf Tfie profit derived from selling a pen is
RM25;find the profit obtained by selling a key
chain, and a calculator
[2001s.1.11]
@All rights reserved
12002.1.41
(s o o\'
(-to c s\
rlo , ol ;p'=11 rs -.r -r4l
'l.-t t 6)
fn o s)
[200151.4]
(t z o\
ttlt
b+c 2c-t)
Find MN and deduce N-l .
Product X, Y and Z are build by combining different
proportion of 3 components A, B, and C. Every
product X, is build by combining 2 components A,
4 components B, and one component C; whereas
for every produc Y, it consists of 3 components A,
3 components B and2 components G; and for
every product Z, itFnsists of 4 components A, I
component B and fi qdmponent C. All together 75O
components A ,1&6 components B and 500
components C are used. By using x, y and z as
the number of components X, Y and Z to be build,
obtains a matrix equation to represnls the
information given.
Hence, find the numberofproducts forX, Y andZ
that have to be built.
[2002K1.10]
\250)
| -l
-1
*=lo 3 rl
[r z 4)
t.lo r ol;r'r-1=-;l o , -tl
'l;ll,) iiT i
I
(-to 4 9\
ir= | rs -4 -r41,
[-s t 6)
(z 3 4\
By using x, y and z stand for the prices for a shirt, a
trouser and a pair of shoe respectively, obtain a
matrix equation to represent thls informatton.
Sotue this matrix equation, and find ihe prices for a
shirt, a trouser, and a p4ir of shoe. [200051.11]
(r o o)
'l
20. MandNarezn\M/
c
Number of trousers [}
Pair of shoes J
Sales orice (RM)
1'
bc
is a symmetry matrixI a = 1, b- 0, c-fra=1, b=1, c =0]. ,
-{tso'"'
B
15;24 l
19. trind the values for a, b, and c such that the matrix
-5
MN and NM.
Hence, find M-r .
During a school holiday season, a mini market offer
3 different packages of sales, that are: A, B, and C
which consists of shirts, trousers, and shoes. The
combination of these items and prices are shown
below:
A
"
it
(-, 2 -l\
z -rl,nnc
lo
s)
,l
l:1
[z z)
Items
l'"'= +ll ; r'j'
t
t)
(-r 2 2\
0 0\
Io
2'l
16.
I tr"
71
Iyl = I lpngl :2oo,so 5ol
l,) [soo
J
The matrix A is given by
(r 2 -3')
a=13 r rl
[o t -2)
(i) Find the matrix B such that B =A2 - 101
,
where I is tne 3 x 3 identity matrix
(ii) Find (A+l)B, and hence find (A + l)''^. B.
(-t r s\ (-t r s\
u -, -ro l; el lzobset sl
-t)
l, -r [, -r .)
1r=l u -2 -,0I' I
22.
Matrix A is given by
(t 3 4'\
A=15 4 tl
[' 23)
13 -
By KKH
\'
STPMSyllabusS&T
Find the adjoiploJ A. Hence, A-'.
[5 marks]
/ro( -r)-r:r
(rzr o o)
t6 -3 -t)
1' \
'l(o:':',:,
o t2t)
rjl-r+V ,, leooor, ot
The matrices P and Q, where PQ = QP, are given
by
/
(2 -2 o\
(-t 1 o\
tttt
P=10 o 21, Q=lo 0 -11
l" o ")
l.t -2 ,)
Determine the values of a, b and c.
Find the rea! numbers m and n for which P = mQ
+ nl, where I is the 3 x 3 idenrity matrix.
[a=0; b=4; c=4; m:2; n=0J[2004P1.91
24.
A, B and C are square .nalrices such that BA = B1
and ABC = (As)r. Show
that
.A_i = 82 = C.
(1 2 o\
tfB=lo -1 ol
[2005P1.8]
['' o 1)
(t o o\
1c=lor ol;
i'r.l -1_to
\
\
,
,
(t ah,
28. Matrix A is siven ov a = | t,/ .rll
I
,ryn/ \-t
(a) Show that 12 = llwhere I is the 3 x 3 identity
matrix, and deduce A-1.
[4 marks]
(b) Find matrix B which satisfies BA =
(: : il
[4 marks][2008P1.4
[-' o 2)
(t o o\ (B -ro 3\
1n-'=lr -r ol
-a'rlr
'1,
t)
lr -2 l" -4 ,)
delerminant of ttre matnx
I
z* +
[o
&,
;1,'
r\''- 1
[20061.3][4 marks]
(s 2 3\
( a r -r8\
zo.6p=lr -+ rl,o=l a -t n I anano
[-13 -l
\
i
0c I
r
t -3
o
\ r-r. t /\t-t, li
: ilL.o
l-o
'' ta j
Determine the values of k such that the
(*
l
{U
o'o
l!ou,
\
.l
,A
\
,)-) q) A=lt-ro ltt-t
[, 2 t)
t3 t 2)
r b ()
[2007s2.e]
A-'-l = i\-' , r Lr
,i r' ,\ ' { $-r r.
c)
=21, where I is the 3 x 3 identity matrix, determine
the values of a, b and c. Hence find P-l. [8 marks]
Two groups of workers have their drinks at a stall.
The first group comprising ten workers have five
cups of tea, two cups of cbffee and three glasses
of fruit juice at a total cost of RM1 1 .80. The
second group of six workers have three cups of
tea, a cup of coffee and two glasses of fruit juice at
a total cost of RM7. 10. The cost of a cup of tea
and ihree glasses of fruit juice is the same as the
cost of four cups of coffee. lf the costs of a cup of
tea, a cup coffee and a glass oi fruit juice are RMx,
RMy and RMz respectively, obtain a matr;x
equation to represent the above information.
Hence, determine the cost of each drink. [6 marks]
la=11, i=-7, c=22: RMl, RM1.30; RM1.40l
,t A = I
h) irft, { i \ } \
\ I;J
\ t-*'
)
s{r\,(_:,iiI,l,"i ,l
5 --(1,
)
"i Lt
4
[20061.111
27. The matrices A and B are given by
(-t
(-ts re 18\
tttt 2 r\
A=l-3 ol,a=l-zt -r3 orl
t0 t' 2)
(-: t2 s)
Find the matrix A2B and deduce the inverse of A.
[5 matus]
Hence,solve the system of linear equations
x-2Y(4 =-8,
3x-tl/42= -15,
@All rights reserved
-14-
By KKH
I
[)
l1