Transcript Matrice

•Addition of Matrices
•Subtraction of Matrices
•Scalar Multiplication of Matrix
•Matrix Multiplication
•Determinant of 2 x 2 matrix
•Inverse Matrix of 2 x 2 matrix
•Solving SLE using Matrix Method
Matrix – rectangular arrangement of variables
or constants into horizontal rows and vertical
columns.
4 columns
Example:
 3 2 1 0 
4

0

2
1


 7 13 2 5 
This would be a 3 x 4
Order – number of rows by number of columns
3 rows
•Row Matrix – only one row
•Column Matrix – only one column
•Square Matrix – same number of rows as
columns
•Equal matrices – same order and each
entry of one matrix is equal to the
corresponding entry of the other matrix.
Addition and Subtraction of Matrices
To add or subtract matrices, they must have the
same order – same number of rows and columns
– because you just add or subtract corresponding
entries.
Example:
 2 3 7   5 6 0 
 9 5 12  7 3 8 

 

7  0   7 9 7 
 2  5 3  6





9


7
5


3

12

8

16
2

4

 

Multiplication of Matrices
Scalar multiplication – multiply the entire
matrix by a number
Example:
 2 9 


3  0
1 
 5 12

 6 27 
 0 3 


 15 36 
Multiplication of Matrices
Matrix multiplication – two matrices can only
be multiplied if the number of columns in the
first equals the number of rows in the second.
2x3 could be multiplied with a 3x4
could not multiply 3x4 and 3x4
The order of the product matrix (what you get after
you multiply) will be the number of rows from the first
and the number of column from the second.
When you multiply the 2x3 and the 3x4, the
product will be a 2x4
Matrix multiplication – to multiply two
matrices, you multiply each row in the first by
each column in the second.
 3 2 
Example:  1
2 0 

3 5 2    0 4 

  1 1


2x3 and 3x2…can multiply and the
product will be a 2x2
 (1)( 3)  (2)(0)  (0)(1)
 
(3)(3)  (5)(0)  (2)(1)
(1)(2)  (2)(4)  (0)(1) 
(3)(2)  ( 5)(4)  (2)(1) 
 3 10 
 


7

12


Example :
A motor manufacturer, with three separate factories, makes two types of
car -one called “standard” and the other called “luxury”.
In order to manufacture each type of car, he needs a certain number of
units of material and a certain number of units of labour each unit
representing £300.
A table of data to represent this information could be
Type
Materials
Labour
Standard
12
15
Luxury
16
20
The manufacturer receives an order from another country to supply
400 standard cars and 900 luxury cars.
He distributes the export order as follows:
Location
Standard
Luxury
Factory A
100
400
Factory B
200
200
Factory C
100
300
The number of units of material and labour needed to complete the order
may be given by the following table:
Location
Materials
Labour
Factory A
100 × 12 + 400 × 16
100 × 15 + 400 × 20
Factory B
200 × 12 + 200 × 16
200 × 15 + 200 × 20
Factory C
100 × 12 + 300 × 16
100 × 15 + 300 × 20
Determinants
Every square matrix has a number associated
with it called a determinant.
Second – order determinant denoted by:
a b
a b 
det 
or

c d
c d 
= ad - bc
Product of the diagonal going down minus
the product of the diagonal going up
Examples:
3 10 
Find det 

4

5


= (3)(-5) – (10)(4)
= -15 – 40
= -55
1 4
Find
3 0
= (1)(0) – (-4)(3)
= 0 – -12
= 12
Identity and Inverse Matrices
Identity matrix is a square matrix that when
multiplied by another matrix, the product
equals that same matrix.
Identity matrix :
 1 0
0 1  ,


1
 1 0 0 
0 1 0  , 0

 0
0 0 1  
0
0
1
0
0
0
0
1
0
0
0
 , etc
0

1 
Identity Matrix has 1
for each element on
the main diagonal and
0 everywhere else.
matrix times inverse = identity matrix
A A  I
1
Not every matrix
has an inverse.
Requirements to have an Inverse
• The matrix must be square (same number of rows
and columns).
• The determinant of the matrix must not be zero.
• A square matrix that has an inverse is called invertible
or non-singular.
• A matrix that does not have an inverse is called
singular.
Inverse of a second order matrix (2 x 2):
a b 
c d 


A
1
1 d

c
det A 
b 

a 
Change the place of a and d and change
the signs of c and b.
Example:
Writing simultaneous equations in matrix form
Consider the simultaneous equations
x + 2y = 4
3x − 5y = 1
In Matrix Form :
Writing
We have AX = B.
This is the matrix form of the simultaneous equations. Here
the unknown is the matrix X,since A and B are already known.
A is called the matrix of coefficients.
Solving the simultaneous equations
Given AX = B, we can multiply both sides by the inverse of A, provided
this exists, to give
A−1AX = A−1B
But A−1A = I, the identity matrix. Furthermore, IX = X, because multiplying
any matrix by an identity matrix of the appropriate size leaves the matrix
unaltered.
So X = A−1B
Example
Solve the simultaneous equations
x + 2y = 4
3x − 5y = 1
Solution
We have already seen these equations in matrix form: