1. 2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
TIF 21101
APPLIED MATH 1
(MATEMATIKA TERAPAN 1)
Week 7
Matrices II
2. 2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
Matrices IIMatrices II
Objectives
Multiplication of Matrices
Identity Matrix
Matrix Inversion
Adjoint of Matrix
Elementary Row Operation
3. 2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
Matrices IIMatrices II
Multiplication between Matrices
It is not same with the scalar multiplication. It
involves multiplication between rows and columns
only.
Rule for this :
You can only multiply two matrices together if the
number of columns of the first equals the number
of rows of the second
4. 2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
Matrices IIMatrices II
5. 2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
Matrices IIMatrices II
Then, [A][B] (read : product of matrices A and B) is given by
AB =
2 x 3 3 x 2
The values must be same
Index of the result
[A]=
2 x 3
[B]=
3 x 2
6. 2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
Matrices IIMatrices II
Let’s take a look its operation…
=
2 x 2
7. 2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
Matrices IIMatrices II
Identity Matrix
The identity matrix of order n is the n x n order of
matrix In = [δij], where δij= 1 if i = j and δij = 0 if i ≠ j.
Therefore:
Multiplying a matrix with its sized identity matrix
will result in the matrix itself.
[A][I] = [I][A] = [A]
8. 2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
Matrices IIMatrices II
Matrix Inversion
The inversion of a matrix is used in devide
operation between matrices.
[A][B]=[C] [B]=[A]-1[C] (Prove it !!!)
[A]-1[A][B]=[A]-1[C]
[I][B]=[A]-1[C]
[B]=[A]-1[C]…it’s proved
9. 2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
Matrices IIMatrices II
If [A] and [B] are square matrices and meet the
condition below:
[A][B]= [B][A]= [I]
then [B] is the invers matrix of [A] and denoted by
[B] = [A] -1.
For order 2 and 3 matrices, its invers matrix can be
found using adjoint method. And for order 3 and
above matrices, we can find using Elementary
Row Operation. It will be discussed later.
10. 2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
Matrices IIMatrices II
Adjoint of Matrix
Suppose [A] is a square matrix and Cij is the
cofactor of [A], then we can reform a new matrix
which contains cij as the elements and then
transpose the new matrix. Thus, it can be called as
adjoint of [A].
Cij = (-1)i+j (Mij)
Mij= Minor element ij
11. 2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
Matrices IIMatrices II
Therefore, to find the invers matrix of [A], we
can use the formula below :
12. 2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
Matrices IIMatrices II
Example :
Find the invers of [A] below :
13. 2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
Matrices IIMatrices II
14. 2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
Matrices IIMatrices II
15. 2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
Matrices IIMatrices II
16. 2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
Matrices IIMatrices II
Elementary Row Operation
This method is used to define some operations
which involve the order 3 and above matrices.
The rules :
1.The interchange between row i with row j, denoted by Rij
2.The multiplication row i with scalar k, denoted by kRi
3.Adding a multiplied row i with scalar k to row j, denoted
by kRi+Rj.
17. 2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
Matrices IIMatrices II
As mention in previous slide, this method can be
applied to find the invers of a matrix.
For example :
Find the invers of
To solve the problem, we need to add some extra
spaces at the right side of the matrix according to
its index.The new spaces will be filled by identity
matrix. Now, our duty is to “move” the right side
into left side and vice versa.
18. 2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
Matrices IIMatrices II
19. 2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
Matrices IIMatrices II
20. 2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
Matrices IIMatrices II
A-1 =
