Section 4-4 Matrices: Basic Operations
Matrix Addition and Subtraction (Prerequisite: The matrices must be of the same size.)
The elements in the sum matrix are found by adding the corresponding elements in the matrix
operands.
1 2 3  2 4 0
Example: 



3 4 4 1  7 5
The elements in the difference matrix are found be subtracting the corresponding elements in
the matrix operands.
1 2 3  2  2 4
Example: 



3 4 4 1    1 3
Scalar Multiplication (Prerequisite: none)
Scalar multiplication means that a matrix is multiplied by a single number (a scalar value). The
resulting matrix is found by multiplying each element of the matrix operand by the scalar
operand.
1 2  5 10 
Example: 5


3 4 15 20
Matrix Multiplication
Multiplying a row matrix by a column matrix.
Prerequisite: The row matrix must have the same number of elements as the column matrix.
The result is a matrix having one row and one column (i.e., a single value). The value is found by
adding the products of the corresponding elements. The elements in a row matrix are
numbered from left to right and the elements in a column matrix are numbered from top to
bottom.
 4
Example: 1 2 35  1  4  2  5  3  6  32
6
Matrix Multiplication in General
Prerequisite: the number of columns in the left operand must be the same as the number of
rows in the right operand. The product matrix will have the same number of rows as the left
operand and the same number of columns as the right operand.
Each element in the product matrix is found by multiplying the corresponding row matrix in the
left operand by the corresponding column matrix in the right operand. That is, the element at
row r and column c in the product is the value found by multiplying row r in the left operand by
column c in the right operand.
1 2 3  2 1  3  2  4 1  (2)  2  1 11 0 
Example: 




3 4 4 1  3  3  4  4 3  (2)  4  1 25  2
Applications
Example 1: A company has two salespersons, Ann and Bob. In August, Ann had $40,000 in sales
and Bob had $35,000 in sales. Ann’s rate of commission is 7% and Bob’s rate of commission is
6.5%. Use matrix arithmetic to find each sales person’s commission and their combined
commissions.
Part One
August Sales : A  40000 35000
0 
0.07
Commission Rates : C1  
0.065
 0
0 
0.07
Commissions  AC1  40000 35000 
 2800 2275
0.065
 0
Part Two
August Sales : A  40000 35000
 0.07 
Commission Rates : C 2  

0.065
 0.07 
TotalCommission  AC 2  40000 35000 
  5075
0.065
Example 2: The September figures were just announced and Ann had $38,500 in sales and Bob
had $41,500 in sales. Use matrix arithmetic to find the total sales for August and September
combined, each sales person’s commission for August and September combined, and their
combined commissions for the two months.
Part One
August Sales : A  40000 35000
September Sales : S  38500 41500
TotalSales  T  A  S  40000 35000  38500 41500  78500 76500
Part Two
0 
0.07
Commission Rates : C1  
0.065
 0
0 
0.07
Commissions  TC1  78500 76500 
 5495 4972.5
0.065
 0
Part Three
0.07
Commission Rates : C 2  

0.65
 0.07 
TotalCommission  TC 2  78500 76500 
  10467.5
0.065
Example 3: A corporation wants to lease a fleet of 12 airplanes with a combined carrying
capacity of 220 passengers. The cost of leasing a 10-passenger airplane is $8,000 a month, the
cost of leasing a 15-passenger airplane is $14,000 a month, and the cost of leasing a 20passenger airplane is $16,000 a month. How many of each type of plane should be leased
assuming the lease cost is to be as small as possible?
Based on our solutions from the previous handout:
0 4 8   8000  $184,000
1 2 9   14000  $180,000

 
 

2 0 10 16000 $176,000
The solution with the smallest lease cost is to lease two 10-passenger airplanes and ten 20passenger airplanes at a cost of $176,000 a month.