# Systems of Equations and Inequalities

## Solving Systems with Inverses


### Learning Objectives

1. Evaluate the determinant of a 2×2 matrix (IA 4.6.1)
2. Evaluate the determinant of a 3x3 matrix (IA 4.6.2)

### Objective 1: Evaluate the determinant of a 2×2 matrix (IA 4.6.1)

If a matrix has the same number of rows and columns, we call it a square matrix. Each square matrix has a real number associated with it called its determinant.


### Practice Makes Perfect


Find the determinant of the 2x2 matrices.

### Objective 2: Evaluate the determinant of a 3×3 matrix (IA 4.6.2)

To evaluate the determinant of a 3×3 matrix, we must be able to evaluate the minor of an entry in the determinant.

The minor of an entry is the 2×2 determinant found by eliminating the row and column in the 3×3 determinant that contains the entry.

For example, to find the minor of entry a1, we eliminate the row and column which contain it. So, we eliminate the first row and first column. Then we write the 2×2 determinant that remains.

To find the minor of entry b2, we eliminate the row and column that contain it. So, we eliminate the second row and second column. Then we write the 2×2 determinant that remains.


### Strategy for evaluating the determinant of a 3x3 matrix

To evaluate a 3×3 determinant we can expand by minors using any row or column. Choosing a row or column other than the first row sometimes makes the work easier.

When we expand by any row or column, we must be careful about the sign of the terms in the expansion. To determine the sign of the terms, we use the following sign pattern chart.

  

### Expanding by minors along the first row to evaluate a 3x3 determinant.

To evaluate a 3×3 determinant by expanding by minors along the first row, we use the following pattern:

NOTE: We can evaluate the determinant of a matrix by expanding minors along any row or column. When a row or a column has a zero entry, expanding by that row or column results in less calculations.


### Practice Makes Perfect

Soriya plans to invest $10,500 into two different bonds to spread out her risk. The first bond has an annual return of 10%, and the second bond has an annual return of 6%. In order to receive an 8.5% return from the two bonds, how much should Soriya invest in each bond? What is the best method to solve this problem?
There are several ways we can solve this problem. As we have seen in previous sections, systems of equations and matrices are useful in solving real-world problems involving finance. After studying this section, we will have the tools to solve the bond problem using the inverse of a matrix.

### Finding the Inverse of a Matrix

We know that the multiplicative inverse of a real number 
       is 
       and 
       For example, 
       and 
       The multiplicative inverse of a matrix is similar in concept, except that the product of matrix 
       and its inverse 
       equals the identity matrix. The identity matrix is a square matrix containing ones down the main diagonal and zeros everywhere else. We identify identity matrices by 
       where 
       represents the dimension of the matrix. Observe the following equations. 
The identity matrix acts as a 1 in matrix algebra. For example, 
      
    
A matrix that has a multiplicative inverse has the properties
A matrix that has a multiplicative inverse is called an invertible matrix. Only a square matrix may have a multiplicative inverse, as the reversibility, 
       is a requirement. Not all square matrices have an inverse, but if 
       is invertible, then 
       is unique. We will look at two methods for finding the inverse of a 
       matrix and a third method that can be used on both 
       and 
  matrices.

### Finding the Multiplicative Inverse Using Matrix Multiplication

We can now determine whether two matrices are inverses, but how would we find the inverse of a given matrix? Since we know that the product of a matrix and its inverse is the identity matrix, we can find the inverse of a matrix by setting up an equation using matrix multiplication.

### Finding the Multiplicative Inverse by Augmenting with the Identity

Another way to find the multiplicative inverse is by augmenting with the identity. When matrix 
         is transformed into 
         the augmented matrix 
         transforms into 
        
      
For example, given
augment 
         with the identity
Perform row operations with the goal of turning 
         into the identity.
1. Switch row 1 and row 2.
         
2. Multiply row 2 by 
  and add to row 1.
        
3. Multiply row 1 by 
  and add to row 2.
        
4. Add row 2 to row 1.
        
5. Multiply row 2 by 
         

         
The matrix we have found is 
           

         

### Finding the Multiplicative Inverse of 2×2 Matrices Using a Formula

When we need to find the multiplicative inverse of a 
         matrix, we can use a special formula instead of using matrix multiplication or augmenting with the identity.
If 
         is a 
         matrix, such as
the multiplicative inverse of 
         is given by the formula
where 
         If 
         then 
         has no inverse.

### Finding the Multiplicative Inverse of 3×3 Matrices

Unfortunately, we do not have a formula similar to the one for a 
         matrix to find the inverse of a 
         matrix. Instead, we will augment the original matrix with the identity matrix and use row operations to obtain the inverse.
Given a 
        
        matrix
augment 
         with the identity matrix
To begin, we write the augmented matrix with the identity on the right and 
         on the left. Performing elementary row operations so that the identity matrix appears on the left, we will obtain the inverse matrix on the right. We will find the inverse of this matrix in the next example.

### Solving a System of Linear Equations Using the Inverse of a Matrix

Solving a system of linear equations using the inverse of a matrix requires the definition of two new matrices: 
       is the matrix representing the variables of the system, and 
       is the matrix representing the constants. Using matrix multiplication, we may define a system of equations with the same number of equations as variables as
To solve a system of linear equations using an inverse matrix, let 
       be the coefficient matrix, let 
       be the variable matrix, and let 
       be the constant matrix. Thus, we want to solve a system 
       For example, look at the following system of equations.
From this system, the coefficient matrix is
The variable matrix is
And the constant matrix is

    Then 
       looks like
Recall the discussion earlier in this section regarding multiplying a real number by its inverse, 
  To solve a single linear equation 
       for 
       we would simply multiply both sides of the equation by the multiplicative inverse (reciprocal) of 
       Thus,
The only difference between a solving a linear equation and a system of equations written in matrix form is that finding the inverse of a matrix is more complicated, and matrix multiplication is a longer process. However, the goal is the same—to isolate the variable.
We will investigate this idea in detail, but it is helpful to begin with a 
  system and then move on to a 
  system.

### Key Equations


### Key Concepts

1. An identity matrix has the property 
       See .
2. An invertible matrix has the property 
       See .
3. Use matrix multiplication and the identity to find the inverse of a 
       matrix. See .
4. The multiplicative inverse can be found using a formula.  See .
5. Another method of finding the inverse is by augmenting with the identity. See .
6. We can augment a 
       matrix with the identity on the right and use row operations to turn the original matrix into the identity, and the matrix on the right becomes the inverse. See .
7. 	Write the system of equations as 
       and multiply both sides by the inverse of 
       See  and .
8. We can also use a calculator to solve a system of equations with matrix inverses. See .

### Section Exercises


### Verbal


### Algebraic

In the following exercises, show that matrix 
         is the inverse of matrix 
        
      
For the following exercises, find the multiplicative inverse of each matrix, if it exists.
For the following exercises, solve the system using the inverse of a 
  matrix.
For the following exercises, solve a system using the inverse of a 
        
        matrix.

### Technology

For the following exercises, use a calculator to solve the system of equations with matrix inverses.

### Extensions

For the following exercises, find the inverse of the given matrix.

### Real-World Applications

For the following exercises, write a system of equations that represents the situation. Then, solve the system using the inverse of a matrix.