FORM 5 ,, TOPIC 4 : MATRICES

mari berkenalan ! 

apa itu metrices ???

4.1 Definition:

A matrix is an ordered set of numbers listed rectangular form.
The ORDER of the Matrix depends on the ROWS and COLUMNS of the Matrix.
Example. Let A denote the matrix

[2 5 7 8]
[5 6 8 9]
[3 9 0 1]
This matrix A has three rows and four columns. We say it is a 3 x 4 matrix. 
The ORDER of the matrix is 3 x 4.

Row matrix
A matrix with one row is called a row matrix

Column matrix
A matrix with one column is called a column matrix 

Square matrix
If a matrix A has n rows and n columns then we say it's a square matrix.

4.2 The sum of matrices of the same kind
Sum of matrices:

To add two matrices of the same kind, we simply add the corresponding elements. 
Examples:
[2 5} + {3 1} = [5 6]


[1 2] + [3 5] = [4 7]
[3 4] . [1 2] . [4 6]

4.3 Scalar multiplication
To multiply a matrix with a real number, we multiply each element with this number. 
Example:

3[1 2] = [3 6]
.[3 4] . [9 12]

4.4 Multiplication of a row matrix by a column matrix
This multiplication is only possible if the row matrix and the column matrix have the same number of elements. The result is a ordinary number ( 1 x 1 matrix).
To multiply the row by the column, one multiplies corresponding elements, then adds the results.
Example.


..........[1]
[2 1 3]. [2] = [19]
..........[5]

Maksudnya: (2x1)+(1x2)+(3x5) = 19

4.5 Multiplication of two matrices A.B
This product is defined only if A is a (p x m) matrix and B is a (m x n) matrix.
So the number of columns of A has to be equal to the number of rows of B.
The product C = A.B then is a (p x n) matrix.
Examples:


[1 2][1 3] = [5 7]
[2 1][2 2] . [4 8]

[1 3][1 2] = [7 5]
[2 2][2 1] . [6 6]

[1 1][2 2] = [0 0]
[1 1][-2 -2] [0 0]

Note: I advise you to find somebody to teach you the concept of multiplication of Matrices if you fail to understand the above solutions.

4.6 INVERSE MATRIX.
Before we can find the inverse of a matrix, we need to first learn how to get the determinant of a matrix. 
Determinant Of A Matrix
If matrix P = [a   b]   then  the determinant is  ad - bc
                    [c   d]

So, the Inverse of P  is  [d  -b]
                                    [-c  a]
                  divided by  ad - bc

cth :

 1.  find the inverse matrix of 

           [ 1     -2  ]

           [ 5      -8 ]

2.  using metrices , calculate the velue of k and  m which  satisfy the following 

matrix equation :

[ 1   -2  ]  [ k ]  =  [ 1 ] 

 [ 5   -8  ]  [ m ] .  [ 11 ]   

solution .... !

1. [ 1   -2 ] = _____1______  [ -8   2  ]

    [ 5   -8  ]    1(-8)-(-2)(5)   [ -5    1 ] 

                 =  _____1______ [-8   2  ]

                         -8+10       [-5   1  ]

=   _1_   [  -8  2  ]

     2     [ -5  1  ]     

2.   [ 1   -2  ]  [ k ]  =  [ 1 ] 

      [ 5   -8  ]  [ m ] .  [ 11 ]   

     [ k ]  =  [  1   -2  ] -1    [  1  ]

      [ m ]    [  5   -8  ]        [ 11  

 [  k  ]   =   _1_  [ -8   2  ]  [ 1 ] 

    [  m  ]        2   [  -5   1  ]  [ 11 ]

 [  k  ]   =   _1_  [ -8 (1) + 2 (11)  ] 

    [  m  ]      2   [  -5(1) + 1(11)   ]

 [  k  ]   =   _1_  [ -8  + 22  ]   =  _1_  [ 14 ]  =  [ 14 bhgi 2 ] 

 [  m  ]      2   [  -5 + 11  ]          2   [  6 ]       [ 6 bhgi 2  ]

k =  7    ,,  m = 3

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