Matrix

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MATRIX





A. DEFINITION OF MATRIX
Matrix is a group of numbers which are arrange based on the row and column. The
arrangement will form the shape of rectangle. The length and the width of a matrix are adjusted
by the number of rows and columns. The number which form the rows and columns of matrix
are called elements of matrix are called elements of matrix.
The notation of matrix are
( )
ij
a
A ,i = 1, 2,…, m and j = 1, 2,…, n
( )
n m 11
11 11 11
11 11 11
11 11 11
n m
a
a a a
a a a
a a a
A
× ×
=
|
|
|
|
|
¹
|





\
|
=
L
M M M M
L
L


Notation of Matrix
A matrix symbolised with block letter, for exmple A , B or C.
Example :
1. A=
|
|
¹
|


\
|
y
x
2. B = (4 – 2 5) 3.C =
|
|
¹
|


\
|
5 4 3
10 8 6

4 . D =
|
|
|
¹
|



\
|

6 4 3 1
0 3 2 1
2 5 4 3


– 1 is element of row 2 and column 1
6 is element of row 3 and column 4





Dimention of Matrix
A matrix with m rows and n columns is called an m-by-n matrix (written m × n) and m and n are
called its dimensions
The dimensions of a matrix are always given with the number of rows first, then the number of
columns
Example :

A =
|
|
¹
|


\
|
5 2 3
4 0 1

The number of row is 2 and the number of column is 3.
Dimention of matrix A is 2 x 3 and writed
3 2×
A


Row 1
Row 2
Row 3
Column 1
Column 3
Column 2
Column 4
Kompetensi dasar : Kemampuan menggunakan sifat-sifat dan operasi matriks untuk
menentukan invers matriks persegi, serta menggunakan determinan dan
invers matriks persegi dalam penyelesaian sistem persamaan linear.
B. Kinds of matrices
o Matrix of row is called matrix in n 1× , example
( ) 5 1 2 X
3 1
− =
×

o Matrix of column is called matrix in 1 m× , example
|
|
|
¹
|



\
|
− =
×
6
1
3
Y
1 3

o Matrix of rectangle is called matrix in m, example
|
|
|
¹
|



\
|
− =
×
7 0 5
1 3 4
2 1 3
H
3 3

o Matrix of triangle is matrix of square in m witch all elements under the main diagonal are
same with zero
|
|
|
¹
|



\
|
=
6 0 0
4 5 0
2 1 3
A
3

o Matrix of lower triangle is matrix of square m which all elements above the main diagonal
are same with zero
|
|
|
¹
|



\
|
=
3 1 2
0 5 4
0 0 3
A
3

o Matrix of identity is matrix of square in m which all main diagonal’s elements are same with
one and the other elements are zero.
|
|
|
¹
|



\
|
=
1 0 0
0 1 0
0 0 1
I
3

o Matrix of zero is matrix in which all of its elements are zero
|
|
|
¹
|



\
|
=
×
0 0 0
0 0 0
0 0 0
I
3 3


C. Matrix of Transpose
Matrix of Transpose is a new matrix which is gained by writing each row of elements in a
first matrix as a column of vice versa.
( ) ( )
m n
ij m n
n m
ij n m
a A a A
×
×
×
×
= ⇒ =
Example :
|
|
|
¹
|



\
|
− = ⇒
|
|
|
¹
|



\
| −
=
3 5 4
7 2 2
3 0 1
A
3 7 3
5 2 0
4 2 1
A
T

B =
|
|
¹
|


\
|
0 9 4
1 7 2
→ B
T
=
|
|
|
¹
|



\
|
0 1
9 7
4 2


D. The Similarity of two matrix
Two matrices A and B are called same, if they have the same size or the same order with the
elements in the same place.
Example :
Known A =
|
|
¹
|


\
|
8 6
1 3
and B =
|
|
¹
|


\
|
x 2 6
1 3
, if A = B then the value of x is ….
answer :

|
|
¹
|


\
|
8 6
1 3
=
|
|
¹
|


\
|
x 2 6
1 3



8 = 2x
x = 4
so the value of x is 2.

For each informations about matrices you can follow this link :
Matrix_(mathematics)
Matrix_2

Exercise 3 . 1
1. Mentioning to the number of line and column from matrices below !
a.
|
|
|
¹
|



\
|
=
9 0
7 5
3 1
A c. P =
|
|
|
¹
|



\
|
z
y
x

b.
|
|
¹
|


\
|

− −
=
9 1 0 5
4 3 2 1
B d. R = ( 3 5 1 6)
2. Determine the dimentions of matricesbelow!
a. A = ( 8 2 0 3 5) c. M =
|
|
|
|
|
¹
|





\
|


5
3
0
1

b.
|
|
¹
|


\
| −
=
8 7 2 0
5 0 1 4
B d. N =
|
|
|
¹
|



\
|
5 0 6
1 0 2
4 5 0





same
same
3. Find the value of x dan y !
a. ( 5x – 2y) = ( 10 4 )
b.
|
|
¹
|


\
|

=
|
|
¹
|


\
|
+ −
+
1
8
2
2
y x
y x

c.
|
|
|
¹
|



\
| −
=
|
|
¹
|


\
|
+ −

2
1
4
1
2 3
4
y x
y x


4. Find the tranpose of matrices below!

a. A =
|
|
¹
|


\
| −
0 2 1
1 4 2
c. C =
|
|
|
|
|
¹
|





\
|


8 2
1 0
6 4
3 5

b. B =
|
|
|
|
|
¹
|





\
|

0
1
2
1
d. D = ( 4 2 5 9 0)
5. Known P =
|
|
¹
|


\
|
− y
x
3
9
and Q =
|
|
¹
|


\
|


4 9
3 5

If
T
P = Q, find the value of x and y!

6. Known
|
|
|
¹
|



\
| −
=
1 3 2
6 0 3
4 1 2
A and
|
|
|
¹
|



\
|
− − =
1 3 4
0 1
2 3 2
p
n
m
B
If
T
A B = , find the value of m , n dan p.


E. Basic Operations
Sum and Reduction of Matrix
Two or more matrices of identical dimensions m and n can be added or reduce, if the
dimenstions are same.
A =
|
|
¹
|


\
|
d c
b a
and B =
|
|
¹
|


\
|
h g
f e

Then A + B =
|
|
¹
|


\
|
d c
b a
+
|
|
¹
|


\
|
h g
f e
=
|
|
¹
|


\
|
+ +
+ +
h d g c
f b e a

A – B =
|
|
¹
|


\
|
d c
b a

|
|
¹
|


\
|
h g
f e
=
|
|
¹
|


\
|
− −
− −
h d g c
f b e a


Example :
1. If P =
|
|
|
¹
|



\
|
3
2
3
and Q =
|
|
|
¹
|



\
|

4
2
0

then P + Q =
|
|
|
¹
|



\
|
3
2
3
+
|
|
|
¹
|



\
|

4
2
0
=
|
|
|
¹
|



\
|
7
0
3
, Q + P =
|
|
|
¹
|



\
|

4
2
0
+
|
|
|
¹
|



\
|
3
2
3
=
|
|
|
¹
|



\
|
7
0
3

because P + Q = Q + P ⇒ comutative
2. If A =
|
|
¹
|


\
|
2 4
1 2
, B =
|
|
¹
|


\
|
3 2
1 0
and C =
|
|
¹
|


\
|
9 8
7 3
, then :
a. ( A + B ) + C =
|
|
¹
|


\
|
+
(
¸
(

¸

|
|
¹
|


\
|
+
|
|
¹
|


\
|
9 8
7 3
3 2
1 0
2 4
1 2

=
|
|
¹
|


\
|
5 6
2 2
+
|
|
¹
|


\
|
9 8
7 3
=
|
|
¹
|


\
|
14 14
9 5

b. A + (B + C) =
|
|
¹
|


\
|
2 4
1 2
+
(
¸
(

¸

|
|
¹
|


\
|
+
|
|
¹
|


\
|
9 8
7 3
3 2
1 0

=
|
|
¹
|


\
|
2 4
1 2
+
|
|
¹
|


\
|
12 10
8 3
=
|
|
¹
|


\
|
14 14
9 5


From example 2a and 2b are assosiative

3. If P =
|
|
¹
|


\
|
2 3
7 4
and Q =
|
|
¹
|


\
|
− 2 3
1 2
, then
a. P – Q =
|
|
¹
|


\
|
2 3
7 4

|
|
¹
|


\
|
− 2 3
1 2

=
|
|
¹
|


\
|
2 3
7 4
+
|
|
¹
|


\
|

− −
2 3
1 2

=
|
|
¹
|


\
|
4 0
6 2

b. Q – P =
|
|
¹
|


\
|
− 2 3
1 2

|
|
¹
|


\
|
2 3
7 4

=
|
|
¹
|


\
|
− 2 3
1 2
+
|
|
¹
|


\
|
− −
− −
2 3
7 4

=
|
|
¹
|


\
|

− −
4 0
6 2

Exercise 3 . 2
1. Simplefy
|
|
¹
|


\
|


+
|
|
¹
|


\
|
− −

8 4 3 5
6 6 6 7
6 3 2 4
2 4 7 6
!
2. If M =
|
|
¹
|


\
|
− 3 4 2
0 3 6
and N =
|
|
¹
|


\
|
− 4 6 3
2 0 1
. Find M + N, N + M, M – N and N – M!
3. Find X from the following equations!

a.
|
|
¹
|


\
| −
= +
|
|
¹
|


\
|


2 0
3 2
6 3
2 4
X
b.
|
|
¹
|


\
|

=
|
|
¹
|


\
|


3 7
1 2
3 5
2 3
X

c.
|
|
¹
|


\
| −
= −
|
|
¹
|


\
|
12 10
16 12
10 12
6 15
X
Scalar and Matrix Multiplication
Given a matrix A and a number c, the scalar multiplication cA is computed by multiplying
every element of A by the scalar c.
A =
|
|
¹
|


\
|
d c
b a
then c.A = p.
|
|
¹
|


\
|
d c
b a
=
|
|
¹
|


\
|
cd cc
cb ca

example :
If
|
|
¹
|


\
|
− −

=
2 5 1
3 2 4
A then
|
|
¹
|


\
|
− −

=
2 5 1
3 2 4
. 4 . 4 A =
|
|
¹
|


\
|
− −

8 20 4
12 8 16

Multiplication of two matrices is well-defined only if the number of columns of the left
matrix is the same as the number of rows of the right matrix.
A
m x n
× B
n x k
= C
m x k



|
|
|
¹
|



\
|
+
+
+
=
|
|
¹
|


\
|

|
|
|
¹
|



\
|
fy ex
dy cx
by ax
y
x
f e
d c
b a

Example :
If
|
|
¹
|


\
|
=
2 4 3
0 1 2
P and
|
|
|
¹
|



\
|
=
3 7
2 6
1 5
Q
Then P × Q =
|
|
¹
|


\
|
2 4 3
0 1 2
|
|
|
¹
|



\
|
×
3 7
2 6
1 5

=
|
|
¹
|


\
|
+ + + +
+ + + +
3 . 2 2 . 4 1 . 3 7 . 2 6 . 4 5 . 3
3 . 0 2 . 1 1 . 2 7 . 0 6 . 1 5 . 2

=
|
|
¹
|


\
|
+ + + +
+ + + +
6 8 3 14 24 15
0 2 2 0 6 10

=
|
|
¹
|


\
|
17 53
4 16


Exercise 3 . 3
1. known p = 3, A =
|
|
¹
|


\
|
4 3
1 2
, B =
|
|
¹
|


\
|
6 5
4 7

Find : a. p. (A.B)
b. (p.A).B
c. (p.B).A

2. If A =
|
|
¹
|


\
|
4
3
, B = (3 1 3) and C =
|
|
|
¹
|



\
|
1 4 5
2 1 0
3 7 4

Find A . (B.C) and (A.B).C

3. If A =
|
|
¹
|


\
|
9 8
6 7
, I =
|
|
¹
|


\
|
1 0
0 1
. Find A.I dan I . A !
4. If A =
|
|
¹
|


\
|
6 5 2
3 2 1
, B =
|
|
|
¹
|



\
|
4 6
8 7
3 1
, C =
|
|
|
¹
|



\
|
0 1
2 3
4 7

Find : a. B+C b. (B+A).A c. C . A d. B.A + C.A

same
F. Determinant of Matrix
1. 2
nd
order of Matrix
If A =
|
|
¹
|


\
|
d c
b a
, then determinant of A is det A or A .
bc ad
d c
b a
A Det − = =
2. 3
rd
order of Matrix
a. Sarrus Methode
|
|
|
¹
|



\
|
=
i h g
f e d
c b a
A
det A =
33 32 31
23 22 21
13 12 11
a a a
a a a
a a a

det A =
32 31 33 32 31
22 21 23 22 21
12 11 13 12 11
a a a a a
a a a a a
a a a a a



det A =
33 21 12 32 23 11 31 22 13 32 21 13 31 23 12 33 22 11
. . . . . . . . . . . . a a a a a a a a a a a a a a a a a a − − − + +

b. Minor – kofaktor
( ) ( ) ( )
in in i2 i2 i1 i1
k a k a k a A det × + + × + × = K
( ) ( ) ( )
nj nj 2j 2j 1j 1j
k a k a k a A det × + + × + × = K
c. Sarrus Methode

( ) ( ) ( ) ( ) ( ) ( ) b d i a f h c e g h d c g f b i e a A det ⋅ ⋅ − ⋅ ⋅ − ⋅ ⋅ − ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅ =

G. The Inverse of Matrix
If A and B are square matrix which same order so that A . B = B . A = I, hence B is inverse
of A and A is inverse of B.
For example :
A =
|
|
¹
|


\
|
2 1
5 3
and B =
|
|
¹
|


\
|


3 1
5 2

then BA =
|
|
¹
|


\
|


3 1
5 2
|
|
¹
|


\
|
2 1
5 3
=
|
|
¹
|


\
|
1 0
0 1
= I
and A . B =
|
|
¹
|


\
|
2 1
5 3
|
|
¹
|


\
|


3 1
5 2
=
|
|
¹
|


\
|
1 0
0 1
= I
Thereby, B is inverse from A, and can be written B = A
– 1
. Because BA = I and B = A
– 1
so
A
– 1
. A = I.


– + – + – +
If
|
|
¹
|


\
|
=
d c
b a
A then A
– 1
bc ab
1

= 0 bc ad ,
a c
b d
≠ −
|
|
¹
|


\
|



Example :
M =
|
|
¹
|


\
|
3 4
1 2
. M
– 1
= …
Answer :
2 4 6 ) 1 )( 4 ( ) 3 )( 2 ( = − = − = − = bc ad D
M
– 1
=
|
|
¹
|


\
|


− a c
b d
bc d a.
1

=
|
|
¹
|


\
|


2 4
1 3
2
1

=
|
|
¹
|


\
|


1 2
2
1
2
3


Exercise 3.4
1. Investigate whether matrix below having invers? If having inverse, determining the
inverse!
a. A =
|
|
¹
|


\
|
4 3
3 2
c. P =
|
|
¹
|


\
|
1 2
2 4

b. B =
|
|
¹
|


\
|


3 1
4 2
d. R =
|
|
¹
|


\
|


1 2
4 8


2. If A =
|
|
¹
|


\
|
6 5
4 2
and B =
|
|
¹
|


\
|
1 2
5 4
, find :
a. ( )
1 −
AB
b.
1 1 − −
A B

3. Find the determinant of matrix below!
a. K =
|
|
|
¹
|



\
|
3 1 2
2 2 1
4 3 2
b. L =
|
|
|
¹
|



\
|

− −
1 1 2
7 5 0
1 4 2


4. If A =
|
|
¹
|


\
|

+
x
x
4 3
2 3
is singular matrix, then the value of x is ….
5. If B =
|
|
¹
|


\
|
+

x
x
4 3
2 3
is singular matrix, then the value of x is ….



H. Application
If A . B = C, so A = C . B
– 1
and B = A
– 1
. C.
The solution from linear equation system
¹
´
¦
= +
= +
q dy cx
p by ax

1.
|
|
¹
|


\
|
=
|
|
¹
|


\
|
|
|
¹
|


\
|
q
p
y
x
d c
b a
and
|
|
¹
|


\
|
|
|
¹
|


\
|



=
|
|
¹
|


\
|
q
p
a c
b d
bc ab
1
y
x

2. D =
q p
b a
= p b q a . . −
Dx =
q r
b c
= r b q c . . −
Dy =
r p
c a
= p c r a . . −
D
Dx
x = and
D
Dy
y =
The solution of linear equation system
¦
¹
¦
´
¦
= + +
= + +
= + +
r iz hy gx
q fz ey dx
p cz by ax

D
D
z ,
D
D
y ,
D
D
x
z
y
x
= = =
With
i h r
f e q
c b p
D
x
= With
r h g
q e d
p b a
D
z
=
With
i r g
f q d
c p a
D
y
= With
i h g
f e d
p b a
D =

Example :
Find the solution of
¹
´
¦
= −
= +
3 5 4
5 2
y x
y x


Answer :
|
|
¹
|


\
|
=
|
|
¹
|


\
|
|
|
¹
|


\
|
− 3
5
5 4
1 2
y
x

|
|
¹
|


\
|
− 5 4
1 2

|
|
¹
|


\
|
y
x
=
|
|
¹
|


\
|
3
5


|
|
¹
|


\
|
y
x
=
|
|
¹
|


\
|

− −
− − 2 4
1 5
4 10
1
|
|
¹
|


\
|
3
5


|
|
¹
|


\
|
y
x
=
|
|
¹
|


\
|


− 14
28
14
1


|
|
¹
|


\
|
y
x
=
|
|
¹
|


\
|
1
2
so x = 2 and y = 1
Exercise 3.5
1. If
|
|
¹
|


\
|
=
1 0
3 2
A ,
|
|
¹
|


\
|
=
3 1
4 2
B and A . M = B, then M is ….
2. Using matrix formula, find the solution of :
a.
¹
´
¦
= +
= −
7 2
4 2 5
y x
y x
d.
¹
´
¦
= − +
= − −
0 14 2
0 13 3 7
y x
y x

b.
¹
´
¦
= + +
= −
0 2 4 3
6 2
y x
y x
e.
¦
¹
¦
´
¦
= + −
= + +
= − −
17 4 3 2
19 3 2 5
11 4 3
z y x
z y x
z y x

c.
¹
´
¦
= −
= +
7 5 4
9 3 2
y x
y x
f.
¦
¹
¦
´
¦
= +
= −
= +
13 2
6 2 2
23 3 2
z y
z x
y x



3. Find matrix X !
a.
|
|
¹
|


\
|
=
|
|
¹
|


\
|
19 17
14 12
4 3
3 2
X
b.
|
|
¹
|


\
|


=
|
|
¹
|


\
|


18 7
13 6
3 1
5 2
. X
c.
|
|
¹
|


\
|
=
|
|
¹
|


\
|
6 12
1 5
6 3
1 2
X

4. Known R =
|
|
|
¹
|



\
| −
c b
a
2 7
4 5
3 2 1
and S =
|
|
|
¹
|



\
| −
b a 3 7 2
3 4 5
3 2 1
, if R = S then c is …
5. Known P =
|
|
|
¹
|



\
| −
1 7
3 3 5
3 2 2
a
c
b
and Q =
|
|
|
¹
|



\
|


1 2 7
3 4 5
3 2 3
b
a
, if P = Q then c is…

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