1. Unit 6 : Matrices

2. What is a Matrix?
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.
This order of this matrix is a 2 x 3.
columns
rows

3. What is the order? (or square matrix) 3 x 3
(Also called a column matrix)
1 x 4
3 x 5
(or square matrix)
2 x 2
4 x 1
(Also called a row matrix)

4. Adding Two Matrices
To add two matrices, they must have the same order. To add, you simply add corresponding entries.

5. = 7 7 4 5 = 7 5 7

6. Subtracting Two Matrices
To subtract two matrices, they must have the same order. You simply subtract corresponding entries.

7. 2-0
-4-1
3-8 -5 2 -5
8-3
0-(-1)
-7-1 5 -8 1 = =
1-(-4)
5-2
0-7 5 3 -7

8. Multiplying a Matrix by a Scalar
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.

9. -3 3 -2 6 -5
-2(-3)
-2(3) 6 -6
-12
-2(6)
-2(-5) 10

10. Multiplication of Matrices
Scalar multiplication – multiply the entire matrix by a number
Example 3:

11. Multiplication of Matrices Matrix multiplication Song
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 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

12. Example 4:
Check :
2x3 and 3x2…can multiply and the product will be a 2x2

13. A table of data to represent this information could be
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
Factory C
300
Using matrix multiplication, find a matrix to represent the number of units of material and labour needed to complete the order.

14. Solution:

15. Determinants
Every square matrix has a number associated with it called a determinant.
Second – order determinant denoted by:
= ad - bc
Product of the diagonal going down minus the product of the diagonal going up
Example 6:
Solution:
Let A =
det A
= (3)(-5) – (10)(4)
= -15 – 40
= -55

16. Identity and Inverse Matrices
Example 7:
Solution:
Let A =
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.
Identity Matrix has 1 for each element on the main diagonal and 0 everywhere else.

17. Inverse of a second order matrix (2 x 2):
matrix times inverse = identity matrix
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):
Change the place of a and d and change the signs of c and b.

18. Example 8:
Find the inverse of
Solution:

19. Solving Simultaneous Equations using inverse matrix
Consider the simultaneous equations
x + 2y = 4
3x − 5y = 1
In Matrix Form :
Let , and
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.