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MTH120 College Algebra Chapter 7.7

 7.7 Solving Systems with Inverses


Solving systems of linear equations using inverses involves finding the inverse of the coefficient matrix and then using it to solve for the vector of variables. Here are the steps for solving systems with inverses and an example:

Step 1: Write the System of Equations

Start with a system of linear equations. For example:

2�+3�=84�−2�=6

Step 2: Formulate the Coefficient Matrix and Constant Vector

Identify the coefficient matrix � and the constant vector �:

Coefficient Matrix �:

�=[234−2]

Constant Vector �:

�=[86]

Step 3: Calculate the Inverse of the Coefficient Matrix

Find the inverse of matrix �, denoted as �−1. If the matrix is invertible, you can use the formula:

�−1=1det(�)⋅adj(�)

Where det(�) is the determinant of �, and adj(�) is the adjugate (adjoint) of �.

Step 4: Solve for the Vector of Variables

Once you have �−1, you can solve for the vector of variables � using the formula:

�=�−1�

Step 5: Interpret the Solution

Interpret the solution in the context of your problem. The values in vector � represent the values of the variables that satisfy the system of equations.

Example:

Let's solve the system of equations:

2�+3�=84�−2�=6

Step 1: Write the System of Equations

The system is already given.

Step 2: Formulate the Coefficient Matrix and Constant Vector

Coefficient Matrix �:

�=[234−2]

Constant Vector �:

�=[86]

Step 3: Calculate the Inverse of the Coefficient Matrix

Find the determinant of matrix �:

det(�)=(2⋅(−2))−(3⋅4)=−4−12=−16

Next, calculate the adjugate of �:

adj(�)=[−2−3−42]

Now, find �−1 using the formula:

�−1=1det(�)⋅adj(�)
�−1=1−16⋅[−2−3−42]=[1831614−18]

Step 4: Solve for the Vector of Variables

Now, solve for the vector of variables � using �−1�:

�=�−1�=[1831614−18][86]=[12]

So, the solution is �=1 and �=2.

This method allows you to solve systems of linear equations by finding the inverse of the coefficient matrix and applying it to the constant vector.


Finding the inverse of a matrix is an important operation in linear algebra, and it's used for various applications, including solving systems of linear equations and transformations. To find the inverse of a square matrix, you can use several methods, including the Gauss-Jordan elimination method, adjugate method, or using software like calculators and computer tools. I'll show you an example of finding the inverse of a matrix using the Gauss-Jordan elimination method:

Example: Finding the Inverse of a 3x3 Matrix

Let's find the inverse of the following 3x3 matrix �:

�=[21−1322103]

Step 1: Augment the Matrix

To find the inverse using the Gauss-Jordan elimination method, create an augmented matrix by appending the identity matrix of the same size to matrix �. The identity matrix for a 3x3 matrix is:

�=[100010001]

Now, form the augmented matrix [�∣�]:

[21−1∣100322∣010103∣001]

Step 2: Apply Row Operations

Use row operations to transform the left side of the augmented matrix into the identity matrix:

  1. Divide Row 1 by 2:
[10.5−0.5∣0.500322∣010103∣001]
  1. Subtract 3 times Row 1 from Row 2:
[10.5−0.5∣0.5000−0.53.5∣−1.510103∣001]
  1. Subtract Row 1 from Row 3:
[10.5−0.5∣0.5000−0.53.5∣−1.5100−0.53.5∣−0.501]
  1. Multiply Row 2 by -2:
[10.5−0.5∣0.50001−7∣3−200−0.53.5∣−0.501]
  1. Add 0.5 times Row 2 to Row 3:
[10.5−0.5∣0.50001−7∣3−20000∣011]

Step 3: Back Substitution

Now that you have transformed the left side into the identity matrix, the right side contains the inverse of matrix �. Perform back substitution:

  1. Subtract 0.5 times Row 3 from Row 1:
[10.5−0.5∣0−0.5−0.501−7∣3−20000∣011]
  1. Add 7 times Row 3 to Row 2:
[10.5−0.5∣0−0.5−0.5010∣357000∣011]
  1. Add 0.5 times Row 2 to Row 1:
[10.50∣1.52.54.5010∣357000∣011]
  1. Subtract 0.5 times Row 1 from Row 2:
[10.50∣1.52.54.5010∣000000∣011]

Step 4: The Inverse

The right side of the augmented matrix is now the inverse of matrix �:

�−1=[1.52.54.5000011]

The inverse is a singular matrix (its determinant is zero), which means the original matrix � is not invertible. In such cases, the system of equations may be linearly dependent, and there may be no unique solution.

The example above demonstrates the Gauss-Jordan elimination method for finding the inverse of a matrix. It's important to note that not all matrices have inverses, and whether an inverse exists depends on the matrix's properties, particularly its determinant.


To find the multiplicative inverse of a square matrix using matrix multiplication, you can follow these steps. Note that not all matrices have multiplicative inverses, and the matrix must be non-singular (i.e., its determinant must be non-zero) for an inverse to exist.

Example: Finding the Multiplicative Inverse of a 2x2 Matrix

Let's find the multiplicative inverse of the following 2x2 matrix �:

�=[2134]

Step 1: Calculate the Determinant of the Matrix

The determinant of matrix � is calculated as:

det(�)=(2⋅4)−(1⋅3)=8−3=5

Since the determinant is non-zero (5≠0), the matrix � is non-singular, and it has a multiplicative inverse.

Step 2: Create the Adjugate Matrix

The adjugate (adjoint) matrix of � is formed by taking the transpose of the cofactor matrix. The cofactor matrix is obtained by replacing each element of � with its corresponding cofactor.

Cofactor of element ���:

  • Remove the �-th row and �-th column of �.
  • Calculate the determinant of the resulting 1x1 matrix.
  • Multiply the determinant by (−1)�+�.

Using these rules, we can calculate the cofactors for �:

  • Cofactor of �11: det=4, (−1)1+1=1, so the cofactor is 4⋅1=4.
  • Cofactor of �12: det=3, (−1)1+2=−1, so the cofactor is 3⋅(−1)=−3.
  • Cofactor of �21: det=1, (−1)2+1=−1, so the cofactor is 1⋅(−1)=−1.
  • Cofactor of �22: det=2, (−1)2+2=1, so the cofactor is 2⋅1=2.

Now, create the cofactor matrix:

Cofactor matrix=[4−3−12]

Transpose the cofactor matrix to obtain the adjugate matrix:

Adjugate matrix=[4−1−32]

Step 3: Calculate the Multiplicative Inverse

The multiplicative inverse (�−1) is found by dividing the adjugate matrix by the determinant of �:

�−1=1det(�)⋅Adjugate(�)=15⋅[4−1−32]

Performing the matrix multiplication:

�−1=15⋅[4−1−32]=[45−15−3525]

So, the multiplicative inverse of matrix � is:

�−1=[45−15−3525]

This is the matrix that, when multiplied by �, yields the identity matrix. In this case, ��−1=�, where � is the identity matrix.


To find the multiplicative inverse of a square matrix by augmenting it with the identity matrix and using row operations, follow these steps. As a reminder, not all matrices have multiplicative inverses, and a matrix must be non-singular (i.e., its determinant is non-zero) for an inverse to exist.

Example: Finding the Multiplicative Inverse of a 2x2 Matrix

Let's find the multiplicative inverse of the following 2x2 matrix �:

�=[2134]

Step 1: Create the Augmented Matrix

To find the inverse using the augmented matrix method, create the augmented matrix [�∣�], where � is the identity matrix of the same size as �:

[�∣�]=[21103401]

Step 2: Perform Row Operations

Use row operations to transform the left side of the augmented matrix into the identity matrix:

  1. Divide Row 1 by 2:
[10.50.503401]
  1. Subtract 3 times Row 1 from Row 2:
[10.50.5002.5−1.51]
  1. Divide Row 2 by 2.5:
[10.50.5001−0.60.4]
  1. Subtract 0.5 times Row 2 from Row 1:
[100.7−0.201−0.60.4]
  1. Subtract 0.7 times Row 2 from Row 1:
[100−0.601−0.60.4]

Step 3: Interpret the Result

The left side of the augmented matrix is now the identity matrix, and the right side contains the inverse of matrix �. In this example, the multiplicative inverse of matrix � is:

�−1=[0−0.6−0.60.4]

This is the matrix that, when multiplied by �, yields the identity matrix. In this case, ��−1=�, where � is the identity matrix.


To find the multiplicative inverse of a 2x2 matrix using a formula, you can use the following general method. For a given 2x2 matrix

�=[����]

the inverse �−1 is given by the formula:

�−1=1��−��[�−�−��]

Let's go through an example:

Example: Finding the Multiplicative Inverse of a 2x2 Matrix

Consider the matrix

�=[2345]

Step 1: Calculate the Determinant

Calculate the determinant of matrix �:

det(�)=(2⋅5)−(3⋅4)=10−12=−2

Step 2: Apply the Formula

Apply the formula to find the inverse:

�−1=1−2[5−3−42]

�−1=[−52322−1]

So, the multiplicative inverse of matrix � is:

�−1=[−52322−1]

You can verify the result by multiplying � with its inverse:

�⋅�−1=[2345]⋅[−52322−1]=[1001]

This should equal the identity matrix, confirming that �−1 is indeed the multiplicative inverse of �.


Finding the multiplicative inverse of a 3x3 matrix involves several steps and a specific formula. Given a 3x3 matrix

�=[�������ℎ�]

The inverse �−1 is given by the formula:

�−1=1det(�)[��−�ℎ�ℎ−����−����−����−����−���ℎ−����−�ℎ��−��]

Here, det(�) is the determinant of matrix �, and �,�,…,� are the elements of the matrix.

Let's go through an example:

Example: Finding the Multiplicative Inverse of a 3x3 Matrix

Consider the matrix

�=[120−1312−14]

Step 1: Calculate the Determinant

Calculate the determinant of matrix �:

det(�)=1(3⋅4−1⋅(−1))−2((−1⋅4)−(2⋅1))+0(2⋅(−1)−(−1⋅2))

det(�)=13

Step 2: Apply the Formula

Apply the formula to find the inverse:

�−1=113[(3⋅4−1⋅(−1))(2⋅1−0⋅(−1))(0⋅(−1)−2⋅3)((−1⋅4)−(2⋅1))(1⋅1−2⋅0)(2⋅(−1)−(−1⋅2))(2⋅(−1)−3⋅0)((−1⋅2)−(2⋅(−1)))(1⋅2−(−1⋅1))]

Simplify the matrix:

�−1=113[132−6−61−4−213]

So, the multiplicative inverse of matrix � is:

�−1=113[132−6−61−4−213]

You can verify the result by multiplying � with its inverse:

�⋅�−1=[120−1312−14]⋅113[132−6−61−4−213]=[100010001]

This should equal the identity matrix, confirming that �−1 is indeed the multiplicative inverse of �

Solving a system of linear equations using the inverse of a matrix is an efficient method, especially when you have multiple equations with the same coefficients. Here's how you can do it with an example:

Example: Solving a System of Linear Equations Using Matrix Inversion

Consider the following system of linear equations:

  1. 2�+3�=11
  2. 4�+5�=19

We can represent this system as a matrix equation ��=�, where:

�=[2345] is the coefficient matrix, �=[��] is the column vector of variables, and �=[1119] is the column vector of constants on the right-hand side.

Step 1: Find the Inverse of the Coefficient Matrix

First, find the inverse of matrix �:

�=[2345]

Calculate the determinant of �:

det(�)=(2⋅5)−(3⋅4)=10−12=−2

Next, find the inverse using the formula for a 2x2 matrix:

�−1=1det(�)[5−3−42]=1−2[5−3−42]=[−52322−1]

Step 2: Solve for � by Matrix Multiplication

Now, you can find the solution vector � by multiplying the inverse �−1 with the constant vector �:

�=�−1�=[−52322−1][1119]=[−52⋅11+32⋅192⋅11−1⋅19]=[−552+57222−19]=[13]

So, the solution to the system of equations is �=1 and �=3.

This method works for systems of linear equations with any number of variables, provided that the coefficient matrix is invertible (i.e., its determinant is non-zero).

To solve a system of linear equations using the inverse of a matrix, you can follow these steps:

Step 1: Set Up the System of Equations

Consider a system of linear equations:

  1. 2�+3�+�=1
  2. 4�+5�+2�=2
  3. 6�+7�+3�=3

You can represent this system in matrix form ��=�, where:

  • � is the coefficient matrix,
  • � is the column vector of variables,
  • � is the column vector of constants on the right-hand side.

Step 2: Create the Coefficient Matrix � and the Constants Vector �

The coefficient matrix � and the constants vector � are:

�=[231452673]

�=[123]

Step 3: Find the Inverse of the Coefficient Matrix �

Calculate the determinant of �:

det(�)=2(5⋅3−2⋅7)−3(4⋅3−2⋅6)+1(4⋅7−5⋅6)=−3

Now, find the inverse of � using the formula for a 3x3 matrix:

�−1=1det(�)adj(�)

Where adj(�) is the adjugate (adjoint) matrix of �. To calculate adj(�), find the cofactor matrix of � by replacing each element with its corresponding cofactor. Then, take the transpose of the cofactor matrix to get adj(�.

After performing these calculations, you find:

�−1=1−3[23−13−4−16825−2−1]

Step 4: Solve for � by Matrix Multiplication

Now, you can find the solution vector � by multiplying the inverse �−1 with the constants vector �:

�=�−1�=1−3[23−13−4−16825−2−1][123]

After performing this multiplication, you'll find the values for �.

In this case, you should get:

�=[2−32]

These values of � are the solution to the system of equations. Each value corresponds to the variable in the same order as in the system, so �=2, �=−3, and �=2.


Here's an example of solving a system of equations using matrix inverses with a calculator:

Example: Solving a System of Equations Using Matrix Inverses

Consider the following system of equations:

  1. 3�+2�=8
  2. 4�−3�=5

We can represent this system as a matrix equation ��=�, where:

�=[324−3] is the coefficient matrix, �=[��] is the column vector of variables, and �=[85] is the column vector of constants on the right-hand side.

Step 1: Calculate the Inverse of the Coefficient Matrix �

Calculate the inverse of matrix �. You can use a calculator that supports matrix operations. Let's denote the inverse as �−1.

�−1=[324−3]−1

Step 2: Find the Solution Vector �

To solve for �, multiply both sides of the equation by �−1:

�=�−1�

Use your calculator to perform the matrix multiplication �−1�. The result will be the solution vector �.

In this example, you should obtain:

�=[23]

So, the solution to the system of equations is �=2 and �=3.

Using a calculator that supports matrix operations can simplify the process, especially for larger systems of equations.

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