matematika terapan week 7

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

TIF 21101

APPLIED MATH 1

(MATEMATIKA TERAPAN 1)

Week 7

Matrices II


Matrices IIMatrices II

Objectives

� Multiplication of Matrices

� Identity Matrix

� Matrix Inversion

� Adjoint of Matrix

� Elementary Row Operation



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



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



Let’s take a look its operation…

=

2 x 2



Identity MatrixThe 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]



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



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.



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



Therefore, to find the invers matrix of [A], we can use the formula below :



Example :

Find the invers of [A] below :



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.



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.



A-1 =

matematika terapan week 7

Education

matrices iimatrices

eng matematika terapan

ilyas hadikusuma

square matrices

product of matrices

invers matrix

square matrix

new matrix