Using an Inverse Matrix to Solve a System of Linear Equations

Instructor: Melanie Olczak

Melanie has taught high school Mathematics courses for the past ten years and has a master's degree in Mathematics Education.

This lesson will show you how to solve a system of linear equations by using inverse matrices. Examples and definitions will be provided to help you understand.

Systems of Equations

Suppose your family is heading to an amusement park. There is one adult and two children and the total cost is $5. Your friend's family joins you. They have three adults and five children and their total cost is $14. How much is each child ticket? How much is each adult ticket?

If you have two unknowns, you need to have two equations to be able to find those unknowns. Having two or more equations with two or more variables is called a system of equations. We can set up a system of equations, using x as the cost of one adult ticket, and y as the cost of one child ticket.


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This system has two equations of lines, and therefore is a system of linear equations. There are many different ways to solve systems of linear equations. We are going to learn how to solve systems of equations by using inverse matrices.

Matrices

A matrix is a rectangular array of numbers enclosed in brackets. In order to solve a system of equations using matrices, we need to create three different kinds of matrices.

The first matrix we will use is called a coefficient matrix, which is just the coefficients, or numbers in front of each variable in the system of equations. We need to be sure that our system of equations is lined up so that each variable is in one column.

If we look at the systems of linear equations example, in the first equation the coefficients are 1 in front of x and 2 in front of y. In the second equation, we have 3 in front of x and 5 in front of y. Since we have two x's and 2 y's, we will have a 2x2 matrix, meaning two rows and two columns. We will call this matrix A.


Coefficient Matrix
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The rows represent the different equations, while the columns represent the different variables. The first row are the coefficients of the first equation and the second row are the coefficients from the second equation. The first column are the coefficients of the x's and the second column are the coefficients of the y's.

The second matrix we need is called the variable matrix. The variable matrix is always going to be one column with the variables in each row. In this case, since we have two variables, we will have a 2x1 matrix, where we have two rows and one column. We will call this matrix X, using a capital letter to differentiate between the lower case x in the problem.


Variable Matrix
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If we look back at the original system of equations, we have used the coefficients and the variables. The only thing left is what the equations are equal to. The third matrix is called the constant matrix which contains the constants of the system of equations. We will call this matrix B. Again this is a 2x1 matrix because it has two rows and one column.


Constant Matrix
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The first equation was equal to 5 so that number goes in the first row. The second equation was equal to 14 so that number goes in the second row.

Once you've changed a system of equations into matrices, how to we solve the system to find the answer?

Using Inverse Matrices to Solve Systems of Equations

Now that we know what matrices we need, we can put them all together to create a matrix equation. A matrix equation contains a coefficient matrix, a variable matrix and a constant matrix, and can be solved. Essential we know that if we multiply matrix A times matrix X it will equal matrix B.


Matrix Equation
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To solve a matrix equation, think about the equation A(X)=B. If we wanted to solve for X, we would need to divide B by A. However, when operating with matrices, we cannot divide. Instead, we will multiply by the inverse of A. We show that we are multiplying by the inverse by using a negative one as an exponent.


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Finding the Inverse of a 2x2 Matrix


Inverse Formula
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In order to find the inverse of a 2x2 matrix, we first switch the values of a and d, second we make b and c negative, finally we multiply by the determinant. The determinant of a matrix is one over the different of ad and bc.

For matrix A, a = 1, b = 2, c = 3 and d = 5. So we plug those values into the inverse formula.


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Now we will simplify. First, we'll simplify the determinant. One time five is five and two times three is six. Five minus six is negative one. One divided by negative one is equal to negative one.


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Next, we multiply all the elements of the matrix by negative one and that gives us the inverse of A.


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