# Systems of Equations and Inequalities

## Matrices and Matrix Operations


### Learning Objectives

1. Write the augmented matrix for a system of equations (IA 4.5.1)
2. Add, subtract matrices and multiply a matrix by a scalar

### Objective 1: Write the augmented matrix for a system of equations (IA 4.5.1)

A matrix is a rectangular array of numbers arranged in rows and columns.
A matrix with m rows and n columns has dimension m×n.

Each number in the matrix is called an element or entry in the matrix.

The matrix on the left below has 2 rows and 3 columns and so it has order 2×3. We say it is a 2 by 3 matrix.

We will use a matrix to represent systems of equations.

Each column then would be the coefficients of one of the variables in the system or the constants.

A vertical line replaces the equal signs.

We call the resulting matrix the augmented matrix for the system of equations.


### Practice Makes Perfect


Write each system of linear equations as an augmented matrix


### Objective 2: Add, subtract matrices and multiply a matrix by a scalar

We add or subtract matrices by adding or subtracting corresponding entries.

In order to do this, the entries must correspond. Therefore, addition and subtraction of matrices is only possible when the matrices have the same dimensions. We can add or subtract a 3 × 3 matrix and another 3 × 3 matrix, but we cannot add or subtract a 2 × 3 matrix and a 3 × 3 matrix because some entries in one matrix will not have a corresponding entry in the other matrix.

The process of scalar multiplication involves multiplying each entry in a matrix by a scalar. A scalar multiple is any entry of a matrix that results from scalar multiplication.


### Practice Makes Perfect 


Perform the indicated operations

Two club soccer teams, the Wildcats and the Mud Cats, are hoping to obtain new equipment for an upcoming season.  shows the needs of both teams.
A goal costs $300; a ball costs $10; and a jersey costs $30. How can we find the total cost for the equipment needed for each team? In this section, we discover a method in which the data in the soccer equipment table can be displayed and used for calculating other information. Then, we will be able to calculate the cost of the equipment.

### Finding the Sum and Difference of Two Matrices

To solve a problem like the one described for the soccer teams, we can use a matrix, which is a rectangular array of numbers. A row in a matrix is a set of numbers that are aligned horizontally. A column in a matrix is a set of numbers that are aligned vertically. Each number is an entry, sometimes called an element, of the matrix. Matrices (plural) are enclosed in [  ] or (  ), and are usually named with capital letters. For example, three matrices named 
 and 
  are shown below.

###  Describing Matrices

A matrix is often referred to by its size or dimensions: 
  indicating 
     rows and 
     columns. Matrix entries are defined first by row and then by column. For example, to locate the entry in matrix 
     identified as 
     we look for the entry in row 
     column 
     In matrix 
     shown below, the entry in row 2, column 3 is 
    
  
A square matrix is a matrix with dimensions 
  meaning that it has the same number of rows as columns. The 
     matrix above is an example of a square matrix.
A row matrix is a matrix consisting of one row with dimensions 
 
  
A column matrix is a matrix consisting of one column with dimensions 
 
  
A matrix may be used to represent a system of equations. In these cases, the numbers represent the coefficients of the variables in the system. Matrices often make solving systems of equations easier because they are not encumbered with variables. We will investigate this idea further in the next section, but first we will look at basic matrix operations.

###  Adding and Subtracting Matrices

We use matrices to list data or to represent systems. Because the entries are numbers, we can perform operations on matrices. We add or subtract matrices by adding or subtracting corresponding entries.
In order to do this, the entries must correspond. Therefore, addition and subtraction of matrices is only possible when the matrices have the same dimensions. We can add or subtract a 
  matrix and another 
  matrix, but we cannot add or subtract a 
  matrix and a 
  matrix because some entries in one matrix will not have a corresponding entry in the other matrix.

### Finding Scalar Multiples of a Matrix

Besides adding and subtracting whole matrices, there are many situations in which we need to multiply a matrix by a constant called a scalar. Recall that a scalar is a real number quantity that has magnitude, but not direction. For example, time, temperature, and distance are scalar quantities. The process of scalar multiplication involves multiplying each entry in a matrix by a scalar. A scalar multiple is any entry of a matrix that results from scalar multiplication.
Consider a real-world scenario in which a university needs to add to its inventory of computers, computer tables, and chairs in two of the campus labs due to increased enrollment. They estimate that 15% more equipment is needed in both labs. The school’s current inventory is displayed in .
Converting the data to a matrix, we have
To calculate how much computer equipment will be needed, we multiply all entries in matrix 
       by 0.15.
We must round up to the next integer, so the amount of new equipment needed is
Adding the two matrices as shown below, we see the new inventory amounts.
This means
Thus, Lab A will have 18 computers, 19 computer tables, and 19 chairs; Lab B will have 32 computers, 40 computer tables, and 40 chairs.

### Finding the Product of Two Matrices

In addition to multiplying a matrix by a scalar, we can multiply two matrices. Finding the product of two matrices is only possible when the inner dimensions are the same, meaning that the number of columns of the first matrix is equal to the number of rows of the second matrix. If 
       is an 
  matrix and 
       is an 
  matrix, then the product matrix 
       is an 
  matrix. For example, the product 
       is possible because the number of columns in 
       is the same as the number of rows in 
       If the inner dimensions do not match, the product is not defined.
We multiply entries of 
       with entries of 
       according to a specific pattern as outlined below. The process of matrix multiplication becomes clearer when working a problem with real numbers.
To obtain the entries in row 
       of 
       we multiply the entries in row 
       of 
       by column 
       in 
       and add. For example, given matrices 
       and 
       where the dimensions of 
       are 
       and the dimensions of 
       are 
       the product of 
       will be a 
       matrix.
Multiply and add as follows to obtain the first entry of the product matrix 
      
    
1. To obtain the entry in row 1, column 1 of 
         multiply the first row in 
         by the first column in 
         and add.
        
2. To obtain the entry in row 1, column 2 of 
         multiply the first row of 
         by the second column in 
         and add.
        
3. To obtain the entry in row 1, column 3 of 
         multiply the first row of 
         by the third column in 
         and add.
        
We proceed the same way to obtain the second row of 
       In other words, row 2 of 
       times column 1 of 
       row 2 of 
       times column 2 of 
       row 2 of 
       times column 3 of 
       When complete, the product matrix will be

### Key Concepts

1. A matrix is a rectangular array of numbers. Entries are arranged in rows and columns.
2. The dimensions of a matrix refer to the number of rows and the number of columns. A 
  matrix has three rows and two columns. See .
3. We add and subtract matrices of equal dimensions by adding and subtracting corresponding entries of each matrix. See , , , and .
4. Scalar multiplication involves multiplying each entry in a matrix by a constant. See .
5. Scalar multiplication is often required before addition or subtraction can occur. See .
6. Multiplying matrices is possible when inner dimensions are the same—the number of columns in the first matrix must match the number of rows in the second.
7. The product of two matrices, 
  and 
  is obtained by multiplying each entry in row 1 of 
  by each entry in column 1 of 
  then multiply each entry of row 1 of 
  by each entry in columns 2 of 
  and so on. See  and .
8. Many real-world problems can often be solved using matrices. See .
9. We can use a calculator to perform matrix operations after saving each matrix as a matrix variable. See .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, use the matrices below and perform the matrix addition or subtraction. Indicate if the operation is undefined.
For the following exercises, use the matrices below to perform scalar multiplication.
For the following exercises, use the matrices below to perform matrix multiplication.
For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed.
For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: 
 
)
For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: 
 
)

### Technology

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. Use a calculator to verify your solution.

### Extensions

For the following exercises, use the matrix below to perform the indicated operation on the given matrix.