Download Keeper 1 - Matrix Operations

GSE Accelerated Pre-Calculus
Keeper 1
A matrix is a rectangular arrangement of numbers in rows and columns.
6 2 −1
−2 0 5
Matrix A has two rows and three columns. The dimensions of this matrix are 2x3
(read “2x3”).
𝐴=
The numbers in a matrix are its entries. In matrix A, the entry in the second row and
third column is 5.
Some matrices have special names because of their dimensions or entries.
Name
Description
Row Matrix
A matrix with only 1
row
Column Matrix
A matrix with only 1
column
Square Matrix
A matrix with the same
number of rows and
columns
Zero Matrix
A matrix whose entries
are all zero
Example
3
−2
0
4
2
1
1
3
−4
−1
0
−3
5
1
6
0
0
0
0
Two matrices are equal if their dimensions are the same and the entries in
corresponding positions are equal.
The following matrices are equal because corresponding entries are equal.
5 0
0
4 3 = 5
−
−1 0.75
4 4
The following matrices are not equal because corresponding entries in the second
row are not equal.
−2 6
−2 6
≠
0 −3
3 0
To add or subtract matrices, you simply add or subtract corresponding entries. You
can add or subtract matrices only if they have the same dimensions.
Examples:
Perform the indicated operation, if possible.
3
1
8 3
2 −7
a. −4 + 0
b.
−
4 0
6 −1
3
7
c.
2 0
1
+
3 4
5
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 a scalar. This process is called
scalar multiplication.
Examples:
Perform the indicated operation(s), if possible.
−2 0
a. 3
4 −7
1
b. −2 0
−4
−2
−4
3 + 6
5
−2
5
−8
6
You can use what you know about matrix operations and matrix equality to solve a
matrix equation.
Example:
Solve the matrix equation for x and y.
4
1
3𝑥 −1
2
+
−2 −𝑦
8
5
=
26
12
0
8
Use matrices to organize the following information about health care plans.
This Year For individuals, Comprehensive, HMO Standard, and HMO Plus cost
$694.32, $451.80, and $489.48, respectively. For families, the Comprehensive, HMO
Standard, and HMO Plus plans cost $1725.36, $1187.76, and $1248.12.
Next Year For individuals, Comprehensive, HMO Standard, and HMO Plus cost
$683.91, $463.10, and $499.27, respectively. For families, the Comprehensive, HMO
Standard, and HMO Plus plans cost $1699.48, $1217.45, and $1273.08.
A company offers the health care plans in the last example to its employees. The
employees receive monthly paychecks from which health care payments are
deducted. Use the matrices in the last example to write a matrix that shows the
monthly changes in health care payments from this year to next year.
 3 4   7 5 
1. 
  10 2 
6
0

 

 6 4   1 3 

2. 


4

5

7
3

 

 2 1 3   3 1 2 

3. 


8 9 6   4 6 7 
 0 5 8   4 1 1
4.  3 3 6    9 5 3 
 4 7 2   5 8 1 
 3 0 
5. 2 

1
2



1 7 
6. 4  3 0 
 1 2 
5 4 2 
7.  3 

0 3 1
10 3 y  10 15
8. 
  6 x 13 
6
13

 

12 3  4 x 3
9. 
   24 5
6
y
5

 

  3x 1  9 4    6 10 
10. Solve  2  
   6 3     4 20  for x and y.
4
y
 
 

