Unit 6 : Matrices. MATRIX: A rectangular arrangement of numbers in rows and columns. The ORDER of a...
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Transcript of Unit 6 : Matrices. MATRIX: A rectangular arrangement of numbers in rows and columns. The ORDER of a...
MATRIX: A rectangular arrangement of numbers in rows and columns.
The ORDER of a matrix is the number of the rows and columns.
The ENTRIES are the numbers in the matrix.
502
126rows
columns
This order of this matrix is a 2 x 3.
67237
89511
36402
3410
200
318 0759
20
11
6
0
7
9
3 x 3
3 x 5
2 x 2 4 x 1
1 x 4
(or square matrix)
(Also called a row matrix)
(or square matrix)
(Also called a column matrix)
To add two matrices, they must have the same order. To add, you simply add corresponding entries.
34
03
12
70
43
35
)3(740
0433
13)2(5
44
40
23
To subtract two matrices, they must have the same order. You simply subtract corresponding entries.
232
451
704
831
605
429
2833)2(1
)4(65015
740249
603
1054
325
In matrix algebra, a real number is often called a SCALAR. To multiply a matrix by a scalar, you multiply each entry in the matrix by that scalar.
14
024
416
08
)1(4)4(4
)0(4)2(4
2 5 3 6 7 0
9 7 5
7 9
3 12
7
16 28 4
Multiplication of Matrices
Scalar multiplication – multiply the entire matrix by a number
Example 3:
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 3x4The dimensions 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.
Row by column, row by column Multiply them line by line Add the products, form a matrix Now you're doing it just fine
Matrix multiplication Song
Example 4:
3 21 2 0
0 43 5 2
1 1
2x3 and 3x2…can multiply and the product will be a 2x2
Check :
(1)( 3) (2)(0) (0)(1) (1)(2) (2)(4) (0)(1)
(3)( 3) ( 5)(0) (2)(1) (3)(2) ( 5)(4) (2)(1)
3 10
7 12
Example 5: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
Using matrix multiplication, find a matrix to represent the number of units of material and labour needed to complete the order.
Solution:
100 40012 15
200 20016 20
100 300
100 12 400 16 100 15 400 20
200 12 200 16 200 15 200 20
100 12 300 16 100 15 300 20
7600 9500
5600 7000
6000 7500
Determinants
Every square matrix has a number associated with it called a determinant.
Second – order determinant denoted by:
deta b a b
orc d c d
= ad - bc
Product of the diagonal going down minus the product of the diagonal going up
Example 6:
3 10det
4 5Find
Solution:
det A= (3)(-5) – (10)(4)= -15 – 40 = -55
3 10
4 5
Let A =
Example 7:
1 43 0
Find
Solution: Let A = 1 4
3 0
det A= (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.
:
1 0 0 01 0 0
1 0 0 1 0 0, 0 1 0 , ,
0 1 0 0 1 00 0 1
0 0 0 1
I dentity matrix
etc
Identity Matrix has 1 for each element on the main diagonal and 0 everywhere else.
matrix times inverse = identity matrix1A A I
Not every matrix has an inverse.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. The determinant of the matrix equal zero.
Inverse of a second order matrix (2 x 2):a b
c d
1 1det
d bA
c aA
Change the place of a and d and change the signs of c and b.
Solving Simultaneous Equations using inverse matrix
Consider the simultaneous equationsx + 2y = 4
3x − 5y = 1
In Matrix Form : 1 2 4
3 5 1
x
y
Let , and1 2
3 5A
xX
y
4
1B
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.