SOLVING SYSTEMS WITH MATRIX EQUATIONS


Unit Overview
In this unit you will explore square matrices, identity matrices and inverse of matrices.  You will then learn how to apply these various features of matrices to solve systems of equations.


Square Matrices and Identity Matrices

A matrix can be used to encode a message and another matrix, it’s inverse, is used to decode a message once it is received.

Square Matrix

A square matrix is a matrix that has the same number of rows and columns.

Example #1:                       

                       2  ×  2                  3  ×  3                         4  ×  4

question1What is true about the number of rows and columns in square matrices?

Square matrices have the same number of rows and columns.

"Click here" to check the answer.


The Identity Matrix for Multiplication


                                        

The product of a real number and 1 is the same number. The product of a square matrix B, and its identity I, is the matrix B.

Example #2:  Multiply Matrix B times the identity matrix of a square matrix.


                              B     ×    I                                                               =      B           

The result of multiplying Matrix B by the identity matrix (Matrix I) is Matrix B.  Just as the product of a real number and 1 is the same number, the product of a square matrix times the identity matrix is the same matrix.

question1If a Matrix C is multiplied by Matrix I (Identity Matrix), what is the result?

Matrix C

"Click here" to check the answer.

Stop!  Go to Questions #1-3 about this section, then return to continue on to the next section.


The Inverse of a Matrix

The product of a real number and its multiplicative inverse is 1.  The product of a square matrix and its inverse is the identity matrix I.

Example #1:  Let C = and  D =

                      Show that C and D are inverses of one another.

To show that Matrix C and Matrix D are inverses of each other, multiply CD and DC.  (Note:  Remember, Matrix multiplication is not always commutative so both CD and DC must be calculated.)

CD = ·                                  DC = ·  

      =                   =

      =                                                            =

The product CD produces the Identity Matrix.  The product DC produces the Identity Matrix.

Since the product of these two matrices CD and DC are equal to the identity matrix, they are inverses of each other.

question1When a matrix is multiplied by its inverse, the result is what kind of matrix?

The Identity Matrix (Matrix I)

"Click here" to check the answer.



Determinant of a 2 × 2 Matrix

Each square matrix can be assigned a real number called the determinant of the matrix


Example #2:  Find the determinant of matrix A, and then determine if matrix A has an inverse.

det(A) = =  ad – bc

       A =

det(A) = ad – bc

det(A) = 7(–1) – 3(2)

det(A) = –13

Since det(A) ≠ 0, matrix A has an inverse.

question1What is true about the determinant of a matrix if the matrix does NOT have an inverse?

The determinant will equal zero.

"Click here" to check the answer.

To use the determinant to find the inverse:

          1.) Find the difference of the cross products.

          2.) Put this number under 1 and multiply it with the matrix using the following changes:
                    a.)  change the location of a and d in the matrix
                    b.)  change the signs of b and c in the matrix


Example #3:  Find the inverse of Matrix B, if it exists.
      
B =                  B −1 =

                                    Note:  ad – bc ≠ 0, so matrix B has an inverse.


                                    B −1 = =

                                    B −1  =


The inverse of    is  


Let's check to see if Matrix A = and Matrix B = are inverses of each other.


question1What matrix is the result of A × B?

AB = the Identity Matrix (I).

"Click here" to check the answer.



question1What matrix is the result of B × A?

BA = the Identity Matrix (I)

"Click here" to check the answer.


question1Are matrices A and B inverses of each other?

Yes

"Click here" to check the answer.

question1  Explain how to determine if matrices A and B are inverses of each other.

Both AB and BA will result in the identity matrix (I).

"Click here" to check the answer.


Stop!  Go to Questions #4-9 about this section, then return to continue on to the next section.


Solving Systems with Matrix Equations

A system of linear equations can be written as a matrix equation.

Example #1:    2a + 4b = –3
                          a  –  b  = 9    

          coefficient                                     variable                                    constant
          matrix, A                                       matrix, X                                 matrix, B

                                                                              

Let's examine how inverses can help solve linear systems written in matrix form.  The following procedure is a general procedure to follow when solving systems of equations using matrices.

       AX = B               *Represent the system as a matrix.
A−1AX = A−1B          *Multiply both sides by the inverse matrix.
        IX = A−1B          * A−1A = I (the Identity Matrix)
         X = A−1B          *IX = X


To solve:
     1.)  find the inverse of the coefficient matrix

                    A =                    A−1 =


                                                                  

     2.)  multiply both sides of the equation by A −1

                             A−1            A         X                    A−1        B
                  ·  ·  =  ·


                                               =                         (A−1 × A = I)

                                                            =                                

a = = 5.5         and         b =

Notice, that on the right side of the equation, we found the solution of the system of equations.  To simplify the procedure, just multiply the Inverse of Matrix A (the coefficient matrix) times Matrix B (the constant matrix).

Quicktime_Video_Icon  Inverse Matrix -- Soccer (03:07)

Let's take a look at another example where we use the simplified method.

Example #2: Solve the following system of equations using matrices:  5x – 4y = 4
                                                                                                                 3x – 2y = 3    

1.)  Determine the coefficient matrix, the variable matrix, and the constant matrix.

          coefficient                                     variable                                    constant
          matrix, A                                       matrix, X                                 matrix, B

                                                                              

Recall that after representing the system as a matrix equation, it can then be solved by multiplying both sides by the inverse matrix to find the solution as shown below algebraically. 

       AX = B               *Represent the system as a matrix.
A−1AX = A−1B          *Multiply both sides by the inverse matrix.
        IX = A−1B          * A−1A = I (the Identity Matrix)
         X = A−1B          *IX = X

We will proceed on determining X = A−1B.

2.)  Find the inverse of the coefficient matrix.


                        A−1 = = 


3.) Multiply the inverse of the coefficient matrix (A−1) times the constant matrix (Matrix B).

                      X = A−1B

                                        A−1       B         
                      = ·  


                           =

                     =
                                         
                 x = 2  and  y = 1.5


Application of Inverse Matrices

Example #3:  Mr. Shriver prepares a 20 question test for his history class. The test has true/false questions worth 3 points each and multiple-choice questions worth 11 points each for a total of 100 points.  Set up a system of equations and use matrices to find the number of each type of question.

     1.)  Define variables to represent the two unknowns.

           Let x = number of true/false questions.
           Let y = number of multiple-choice questions.

     2.)  Write a system of equations for the problem.

x + y = 20    There are x amount of true/false questions and y amount of multiple-choice questions with a total of 20 questions on the test.
   
3x + 11y = 100 The value of the true/false questions is 3x and the value of the multiple choice questions is 11x with a total value of 100 points on the test. 

 
The system of equations is:

  x  +     y = 20 *The coefficients of x and y are "understood" to be 1.
(1x + 1y = 20) 
3x  + 11y = 100  

3.)  Write the system of equations as a matrix equation.  (Note:  Matrix A is the coefficient matrix and Matrix B is the constant matrix.

A =                          B =

The Matrix Equation is .

4.)  Find the inverse of the coefficient matrix (Matrix A) where a = 1, b = 1, c = 3,
and d = 11.

 A−1 = =

5.)  Solve the matrix equation by multiplying the inverse of Matrix A times the constant matrix, Matrix B.

            A −1   ·     B

     ·

       =       =


There are 15 true/false questions and 5 multiple-choice questions on the test.




Stop!  Go to Questions #10-20 to complete this unit.




Below are additional educational resources and activities for this unit.
 
Click on the icon to find and practice topics for this unit.
 
Determinants of 2x2 Matrices
 
Matrix Inverses