Matrices The Basics
Vocabulary and basic concepts
A matrix is an array of numbers
For example
Three classes are split into boys and girls.
A: 15 boys and 17 girls.
B: 18 boys and 14 girls.
C: 10 boys and 19 girls.
This data could be arranged as a 2 X 3
MATRIX.
 15 18 10 


17 14 19 
Mrs. S Richards
A matrix with R rows and C
columns is called an
R X C Matrix.
This is called the ORDER of the
matrix.
Each entry in the matrix is called
an ELEMENT.
Mrs. S Richards
The CO-ORDINATE (2,3) has ONE ROW and TWO
COLUMMS. It is a 1X2 matrix
A matrix with only one row is called a
ROW VECTOR.
6
 
 
The vector  2  represents a TRANSLATION or shift
6 units to the right and 2 units down.
A matrix with only one column and any number of
rows is called a
COLUMN VECTOR.
A Matrix with the same number of
ROWS as COLUMNS is called a
SQUARE MATRIX.
Mrs. S Richards
1 2 6


6
3
7


 3 0 10 


Addition and Subtraction of Matrices
We may add or subtract two matrices ONLY if they have the
SAME ORDER.
Adding or subtracting matrices is carried out by adding or
subtracting their corresponding elements.
 2 1   6 5   8 4 




3

4
1

7
4

11

 
 

 2 3 4   1 6 2   1 3 2 

 
 

1

3
6

0

5
4

1
2
2

 
 

 5 7 0   2 5 3   3 2 3 

 
 

If two matrices A and B are of UNEQUAL ORDER
then A + BMrs.does
not exist.
S Richards
Addition and Subtraction of Matrices
If two matrices A and B are of EQUAL ORDER then
A+B=B+A
Addition of Matrices (where allowed) is COMMUTATIVE
 2 1   6 5   6 5   2 1 





3

4
1

7
1

7
3

4

 
 
 

 8 4 


4

1
1


Mrs. S Richards
EQUAL MATRICES
Two matrices A and B are equal if EVERY element in A
is equal to EVERY element in B.
Two matrices may be the SAME ORDER but NOT EQUAL.
 4 3  4 3



 1 0   1 0 
Mrs. S Richards
CAN YOU
SEE WHY?
Multiplication by a Scalar
If k is a constant scalar, the matrix kA
is such that EACH OF ITS ENTRIES ARE
MULTIPLIED by k
 4 3
A

 1 0 
 8 6
2A  

 2 0 
 4k 3k 
kA  

 1k 0k 
Mrs. S Richards
1
5
 54 53 
A   1

 5 0
The ZERO or NULL Matrix
Has every element ZERO.
0 0


0 0
0 0 0


0 0 0
Adding or subtracting the NULL Matrix will
have no effect.
MULTIPLYING by the NULL Matrix will create
another NULL MATRIX.
Mrs. S Richards
The UNIT or IDENTITY MATRIX I
 1 0  A VERY IMPORTANT MATRIX. (Later work!)


1 0 0
These are always SQUARE
0 1


0
1
0


0 0 1


MATRICES
This is denoted by the letter I and has zero entries except
for 1’s on the diagonal.
MULTIPLYING by the IDENTITY Matrix will create the SAME
MATRIX. In a similar way to Multiplying a number by 1
Mrs. S Richards
The TRANSPOSE OF A MATRIX
The rows are written as columns and the
columns as rows.
 1 2 
A= 

0 5 
 3 1 2 


B  1 2 0
 4 5 7 


 1 0
A 

 2 5 
t
3 1 4


Bt   1 2 5 
2 0 7


Mrs. S Richards