Download basic matrix operations

7.2 BASIC MATRIX OPERATIONS
Matrices are useful tools in helping organize data.
Example 1. The EZ Life Company manufactures sofas and armchairs in three models:
A, B, and C. The company has regional warehouses in New York, Chicago and San
Francisco. In its August shipment, the company sends 10 model A sofas, 12 model B sofas,
5 model C sofas, 15 model A chairs, 20 model B chairs, and 8 model C chairs to each
warehouse.
This data might be organized by first listing it as follows:
Sofas
Chairs
10 model A
15 model A
12 model B
20 model B
5 model C
8 model C
Alternatively, we might tabulate the data in a chart.
Sofa
Chair
M O D E L
A
B
10
12
15
20
C
5
8
With the understanding that the numbers in each row refer to the furniture type (sofa, chair)
and the numbers in each column refer to the model (A, B, C), the same information can be
given a matrix, as follows:
10 12 5
M 
.
15 20 8
This is a 2 3 matrix. A matrix with m rows and n columns has dimensions or size m n .
The number of rows is always given first.
A matrix with only one row is called a row matrix or row vector. A matrix with only one
column is called a column matrix or column vector. A matrix with the same number of
rows as columns is called a square matrix.
When a matrix is denoted by a single letter, such as matrix M above, then the element in
row i and column j is denoted ai j . For example, a 2 1 = 15 (the element in row 2, column 1).
Similarly, a1 3 = 5 (the element in row 1, column 3).
ADDITION AND SUBTRACTION
The sum of two m n matrices X and Y is the m n matrix X  Y in which
each element is the sum of the corresponding elements of X and Y.
It is important to remember that only matrices of the same size can be added.
Example 2. Find each sum, if possible.
a)
5  6  4 6 
8 9    8  3

 

b)
 5 6   3 9 1
6 2  4 2 5

 

We can also subtract two matrices.
If X and Y are two matrices of the same size, then
X  Y  X   Y 
Example 3. Find the difference, if possible:
a)
5 6   4 6 
8 9    8 3

 

b)
5 6   3 9 1
 6 2    4 2 5

 

Two matrices are equal if they are the same size and if corresponding elements, position by
position, are equal.
Example 4. Find the values of the variables that make each statement true, if possible.
a)
 2 1  x y
 p q    1 0 

 

b)
1 
x  
 y   4
  0
 
SCALAR MULTIPLICATION
In work with matrices, a real number is called a scalar to distinguish it from a matrix. The
product of a scalar k and a matrix X is the matrix kX, where each element of X is multiplied
by the scalar k.
Example 5. Find the product.
1 
5  4 
 0 
 4 6 
 1 0
Example 6. Let A  
and B  

 . Find 2 A  3B .
 8 3
 2 4
MATRIX EQUATIONS
Example 7. Solve for X in the equation 3X  A  B , where
1  2
  3 4
and
A
B


 2 1
0 3 


MATRIX MULTIPLICATION
We do not multiply two matrices by multiplying corresponding entries. Instead, we must
think of matrix multiplication as row-by-column multiplication. To better understand how
this works, let’s begin with the definition of matrix multiplication for matrices of order
2 2 .
Definition of Matrix Multiplication for 2  2 Matrices:
a b   e
AB  

c d   g
f   ae  bg

h   ce  dg
af  bh 
cf  dh 
Here’s how to think of this multiplication:
a b  e

 g


a b 





 e
c d   g




c d  


  ae  bg

 
f 

h  
 
  ce  dg
 
f 

h  



af  bh 






cf  dh 
In order to multiply two matrices, the number of columns of the first matrix must be
equal to the number of rows of the second matrix.
MATRIX A
MATRIX B
m n
n p
These must be equal
The order of AB is
m p
 2 3
0 1 
Example 7. Find AB, given that A  
and B  


4 7
5 6 
Example 8.
4
Given A  1 2 3 and B   5  . Find AB and BA.
 6 
Example 9. Find each product, where possible.
a)
4 2  1 2 3 4
1 3  0 2 1 6 



b)
 1 2 3 4 4 2
0 2 1 6  1 3 



7.4
THE DETERMINANT OF A MATRIX
Every n  n matrix A is associated with a real number called the determinant of a matrix A
and is written as A . The determinant of a 2  2 matrix is defined below:
a b
a b 
If A  
,
then
A

 ad  bc .

c d
c d 
Note: Matrices are enclosed with square brackets, while determinants are denoted
with vertical bars. Also, a matrix is an array of numbers, but a determinant is a
single number.
 3 4
Example 10. Let A  
 . Find A .
 6 8
To evaluate a 3  3 determinant, we will use a process called expansion by minors. The
minor or an element is the determinant that remains after deleting the row and column of
that element.
Example 11. Find the minor of the following elements of the matrix
6 2 4 
8 9 3 


1 2 0 
a) 6
b) 3
c) 8
To expand by minors about any row or column, use alternating plus and minus signs to
precede the numerical factors of the minors according to the following array of signs:

 
 


 
6 2 4
Example 12. Evaluate:
a)
8 9 3
9
b)
1 2 0
5
0
2 3 0
1
4
2
Example 13. Find the value of x:
a)
4x x
9
x
 28
b)
x 1 1
2
x
0