PrecalculusMathematics and Statistics
Solving Systems with Inverses
Nancy 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 Nancy 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 numberisandFor example, andThe multiplicative inverse of a matrix is similar in concept, except that the product of matrixand its inverseequals the identity matrix. The identity matrix is a square matrix containing ones down the main diagonal and zeros everywhere else. We identify identity matrices bywhererepresents the dimension of the matrix. [link] and [link] are the identity matrices for amatrix and amatrix, respectively.
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 ifis invertible, thenis unique. We will look at two methods for finding the inverse of amatrix and a third method that can be used on bothandmatrices.
The identity matrix,is a square matrix containing ones down the main diagonal and zeros everywhere else.
Ifis an matrix andis an matrix such thatthenthe multiplicative inverse of a matrix
Given matrix A, show that
Use matrix multiplication to show that the product ofand the identity is equal to the product of the identity and A.
Given two matrices, show that one is the multiplicative inverse of the other.
- Given matrixof orderand matrixof ordermultiply
- Ifthen find the productIfthenand
Show that the given matrices are multiplicative inverses of each other.
MultiplyandIf both products equal the identity, then the two matrices are inverses of each other.
andare inverses of each other.
Show that the following two matrices are inverses of each other.
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.
Use matrix multiplication to find the inverse of the given matrix.
For this method, we multiplyby a matrix containing unknown constants and set it equal to the identity.
Find the product of the two matrices on the left side of the equal sign.
Next, set up a system of equations with the entry in row 1, column 1 of the new matrix equal to the first entry of the identity, 1. Set the entry in row 2, column 1 of the new matrix equal to the corresponding entry of the identity, which is 0.
Using row operations, multiply and add as follows:Add the equations, and solve for
Back-substitute to solve for
Write another system of equations setting the entry in row 1, column 2 of the new matrix equal to the corresponding entry of the identity, 0. Set the entry in row 2, column 2 equal to the corresponding entry of the identity.
Using row operations, multiply and add as follows:Add the two equations and solve for
Once more, back-substitute and solve for
Finding the Multiplicative Inverse by Augmenting with the Identity
Another way to find the multiplicative inverse is by augmenting with the identity. When matrixis transformed intothe augmented matrixtransforms into
For example, given
augmentwith the identity
Perform row operations with the goal of turninginto the identity.
- Switch row 1 and row 2.
- Multiply row 2 byand add to row 1.
- Multiply row 1 byand add to row 2.
- Add row 2 to row 1.
- 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 amatrix, we can use a special formula instead of using matrix multiplication or augmenting with the identity.
Ifis amatrix, such as
the multiplicative inverse ofis given by the formula
whereIfthenhas no inverse.
Use the formula to find the multiplicative inverse of
Using the formula, we have
Use the formula to find the inverse of matrixVerify your answer by augmenting with the identity matrix.
Find the inverse, if it exists, of the given matrix.
We will use the method of augmenting with the identity.
- Switch row 1 and row 2.
- Multiply row 1 by −3 and add it to row 2.
- There is nothing further we can do. The zeros in row 2 indicate that this matrix has no inverse.
Finding the Multiplicative Inverse of 3×3 Matrices
Unfortunately, we do not have a formula similar to the one for amatrix to find the inverse of amatrix. Instead, we will augment the original matrix with the identity matrix and use row operations to obtain the inverse.
Given a matrix
augmentwith the identity matrix
To begin, we write the augmented matrix with the identity on the right andon 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.
Given amatrix, find the inverse
- Write the original matrix augmented with the identity matrix on the right.
- Use elementary row operations so that the identity appears on the left.
- What is obtained on the right is the inverse of the original matrix.
- Use matrix multiplication to show thatand
Given thematrixfind the inverse.
Augmentwith the identity matrix, and then begin row operations until the identity matrix replacesThe matrix on the right will be the inverse of
Find the inverse of thematrix.
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, andis 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, letbe the coefficient matrix, letbe the variable matrix, and letbe the constant matrix. Thus, we want to solve a systemFor example, look at the following system of equations.
From this system, the coefficient matrix is
The variable matrix is
And the constant matrix is
Recall the discussion earlier in this section regarding multiplying a real number by its inverse,To solve a single linear equationforwe would simply multiply both sides of the equation by the multiplicative inverse (reciprocal) ofThus,
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 asystem and then move on to asystem.
Given a system of equations, write the coefficient matrixthe variable matrixand the constant matrixThen
Multiply both sides by the inverse ofto obtain the solution.
If the coefficient matrix does not have an inverse, does that mean the system has no solution?
No, if the coefficient matrix is not invertible, the system could be inconsistent and have no solution, or be dependent and have infinitely many solutions.
Solve the given system of equations using the inverse of a matrix.
Write the system in terms of a coefficient matrix, a variable matrix, and a constant matrix.
First, we need to calculateUsing the formula to calculate the inverse of a 2 by 2 matrix, we have:
Now we are ready to solve. Multiply both sides of the equation by
The solution is
Can we solve forby finding the product
No, recall that matrix multiplication is not commutative, soConsider our steps for solving the matrix equation.
Notice in the first step we multiplied both sides of the equation bybut thewas to the left ofon the left side and to the left ofon the right side. Because matrix multiplication is not commutative, order matters.
Solve the following system using the inverse of a matrix.
Write the equation
First, we will find the inverse ofby augmenting with the identity.
Multiply row 1 by
Multiply row 1 by 4 and add to row 2.
Add row 1 to row 3.
Multiply row 2 by −3 and add to row 1.
Multiply row 3 by 5.
Multiply row 3 byand add to row 1.
Multiply row 3 byand add to row 2.
Multiply both sides of the equation byWe want
The solution is
Solve the system using the inverse of the coefficient matrix.
Given a system of equations, solve with matrix inverses using a calculator.
- Save the coefficient matrix and the constant matrix as matrix variablesand
- Enter the multiplication into the calculator, calling up each matrix variable as needed.
- If the coefficient matrix is invertible, the calculator will present the solution matrix; if the coefficient matrix is not invertible, the calculator will present an error message.
Solve the system of equations with matrix inverses using a calculator
On the matrix page of the calculator, enter the coefficient matrix as the matrix variableand enter the constant matrix as the matrix variable
On the home screen of the calculator, type in the multiplication to solve forcalling up each matrix variable as needed.
Evaluate the expression.
Access these online resources for additional instruction and practice with solving systems with inverses.
|Identity matrix for amatrix|
|Identity matrix for amatrix|
|Multiplicative inverse of amatrix|
- An identity matrix has the propertySee [link].
- An invertible matrix has the propertySee [link].
- Use matrix multiplication and the identity to find the inverse of amatrix. See [link].
- The multiplicative inverse can be found using a formula. See [link].
- Another method of finding the inverse is by augmenting with the identity. See [link].
- We can augment amatrix 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 [link].
- Write the system of equations asand multiply both sides by the inverse ofSee [link] and [link].
- We can also use a calculator to solve a system of equations with matrix inverses. See [link].
In a previous section, we showed that matrix multiplication is not commutative, that is,in most cases. Can you explain why matrix multiplication is commutative for matrix inverses, that is,
Ifis the inverse ofthenthe identity matrix. Sinceis also the inverse ofYou can also check by proving this for amatrix.
Does everymatrix have an inverse? Explain why or why not. Explain what condition is necessary for an inverse to exist.
Can you explain whether amatrix with an entire row of zeros can have an inverse?
No, becauseandare both 0, sowhich requires us to divide by 0 in the formula.
Can a matrix with an entire column of zeros have an inverse? Explain why or why not.
Can a matrix with zeros on the diagonal have an inverse? If so, find an example. If not, prove why not. For simplicity, assume amatrix.
Yes. Consider the matrixThe inverse is found with the following calculation:
In the following exercises, show that matrixis the inverse of matrix
For the following exercises, find the multiplicative inverse of each matrix, if it exists.
There is no inverse
For the following exercises, solve the system using the inverse of amatrix.
For the following exercises, solve a system using the inverse of a matrix.
For the following exercises, use a calculator to solve the system of equations with matrix inverses.
For the following exercises, find the inverse of the given matrix.
For the following exercises, write a system of equations that represents the situation. Then, solve the system using the inverse of a matrix.
2,400 tickets were sold for a basketball game. If the prices for floor 1 and floor 2 were different, and the total amount of money brought in is $64,000, how much was the price of each ticket?
In the previous exercise, if you were told there were 400 more tickets sold for floor 2 than floor 1, how much was the price of each ticket?
A food drive collected two different types of canned goods, green beans and kidney beans. The total number of collected cans was 350 and the total weight of all donated food was 348 lb, 12 oz. If the green bean cans weigh 2 oz less than the kidney bean cans, how many of each can was donated?
Students were asked to bring their favorite fruit to class. 95% of the fruits consisted of banana, apple, and oranges. If oranges were twice as popular as bananas, and apples were 5% less popular than bananas, what are the percentages of each individual fruit?
50% oranges, 25% bananas, 20% apples
A sorority held a bake sale to raise money and sold brownies and chocolate chip cookies. They priced the brownies at $1 and the chocolate chip cookies at $0.75. They raised $700 and sold 850 items. How many brownies and how many cookies were sold?
A clothing store needs to order new inventory. It has three different types of hats for sale: straw hats, beanies, and cowboy hats. The straw hat is priced at $13.99, the beanie at $7.99, and the cowboy hat at $14.49. If 100 hats were sold this past quarter, $1,119 was taken in by sales, and the amount of beanies sold was 10 more than cowboy hats, how many of each should the clothing store order to replace those already sold?
10 straw hats, 50 beanies, 40 cowboy hats
Anna, Ashley, and Andrea weigh a combined 370 lb. If Andrea weighs 20 lb more than Ashley, and Anna weighs 1.5 times as much as Ashley, how much does each girl weigh?
Three roommates shared a package of 12 ice cream bars, but no one remembers who ate how many. If Tom ate twice as many ice cream bars as Joe, and Albert ate three less than Tom, how many ice cream bars did each roommate eat?
Tom ate 6, Joe ate 3, and Albert ate 3.
A farmer constructed a chicken coop out of chicken wire, wood, and plywood. The chicken wire cost $2 per square foot, the wood $10 per square foot, and the plywood $5 per square foot. The farmer spent a total of $51, and the total amount of materials used wasHe usedmore chicken wire than plywood. How much of each material in did the farmer use?
Jay has lemon, orange, and pomegranate trees in his backyard. An orange weighs 8 oz, a lemon 5 oz, and a pomegranate 11 oz. Jay picked 142 pieces of fruit weighing a total of 70 lb, 10 oz. He picked 15.5 times more oranges than pomegranates. How many of each fruit did Jay pick?
124 oranges, 10 lemons, 8 pomegranates
- Linear Functions
- Polynomial and Rational Functions
- Exponential and Logarithmic Functions
- Trigonometric Functions
- Periodic Functions
- Trigonometric Identities and Equations
- Further Applications of Trigonometry
- Systems of Equations and Inequalities
- Introduction to Systems of Equations and Inequalities
- Systems of Linear Equations: Two Variables
- Systems of Linear Equations: Three Variables
- Systems of Nonlinear Equations and Inequalities: Two Variables
- Partial Fractions
- Matrices and Matrix Operations
- Solving Systems with Gaussian Elimination
- Solving Systems with Inverses
- Solving Systems with Cramer's Rule
- Analytic Geometry
- Sequences, Probability and Counting Theory
- Introduction to Calculus