Lesson 5

Inverse of a Square Matrix

Objective: Use the inverse of a square matrix to solve linear systems

Reading Assignment:  Pages 159 through 169

Class Notes:
Identity Matrix
"1" is the identity for multiplication, when you multiply a number by 1, the product is the same as the number.  Matrix multiplication also has an identity matrix, I.  The identity matrix is a square matrix (same number of rows and columns) with 1's in the principal diagonal and 0's everywhere else in the matrix.  The 2 x 2 identity matrix is

Example 1:

Solution:
Using the calculator,  

Inverse of a Square Matrix
 is the multiplicative inverse for the number 2 (the product of a number and its multiplicative inverse equals the multiplicative identity, 1).  Matrix multiplication also has an identity matrix, the product of a matrix A and it inverse equals the identity matrix, I.

Example 2:
Show that is the inverse matrix for the square matrix
Solution:
For square matrices,  .  Using the Calculator to multiply the matrices produces

Finding the inverse of a square matrix
There is an algorithm using row operations to find the inverse of a square matrix.  This procedure can become very tedious for large matrices.  Luckily, the calculate will find the inverse of a square matrix.

Example 3:
Find the inverse for the matrix

Solution:

Using the TI-83 to Find the Inverse of a Square Matrix
1.  Enter the matrix into the calculator.
     A.  Access the Matrix menu by pressing [2nd] and [].
     B.  Use the right arrow key to highlight "EDIT" and press [ENTER].
           (The matrix will be stored in matrix A.)
     C.  Enter the dimensions of the matrix.  In this case, type in 3, [ENTER], and 3,
           [ENTER].
     D.  Type in each element of the matrix, pressing [ENTER] after each value.
     E.  Exit the Matrix Menu by pressing [2nd] and [MODE].
3.  To find the inverse matrix:
     A.  Access the Matrix menu by pressing [2nd] and [].
     B.  Press [ENTER] to access matrix A.
     C.  Press the key.
     D.  Press [ENTER] to find the inverse matrix

              Note:  Since the decimals in the inverse matrix are terminating, this matrix can be
          used as the inverse.  However, if the decimals were nonterminating, then the 
          elements in the inverse matrix should be changed to fractions.
          i.  Press the [MATH] button.
          ii.  From the MATH window, "1: Frac" is highlighted. Press [ENTER].
          iii. "Ans>Frac" appears on the screen.  By pressing [ENTER], the matrix in 
              memory will be converted to fractions    

Matrix Equations
A matrix equation is comprised of a square matrix (n x n) A, a column matrix (n x 1) X, and a column matrix (n x 1) B.  These matrices are combined to form AX = B.  To solve this matrix equation, multiply both sides of the equation by .  Then X = (B).

Starting with:           AX = B
                     (AX) = (B)
                     (A)X = (B)                           Since (A) = I, the identity matrix
                               IX =  (B)                           Since IX = X
                                X = (B)  

Example 4:
Use matrix equations and the inverse matrix to solve the linear system:
  x + 2y + 3z =   4
          y +   z = -2
2x + 2y         =  3
Solution:
Set up matrixes A, X, B

Use the calculator to find .      

Store the inverse matrix in matrix D.  
1.  Press "STO:"
2.  Access the Matrix menu by pressing [2nd] and [].
3.  Use the down arrow to highlight "4: [D]" and press [ENTER].
4.  Press [ENTER].  (The inverse will be stored in matrix D.)

Use matrix multiplication to solve X = (B) .  
x = 5.75,  y = -4.25, and z = 2.25

Testing the solution:   x + 2y + 3z =  5.75 + 2(-4.25) + 3(2.25) = 4.75 - 8.5 + 6.75 = 4
                                        y +   z = -4.25 + 2.25 = 2
                               2x + 2y        =  2(5.75) + 2(-4.25) = 11.5 - 8.5 = 3

Example 5:
 Use matrix equations and the inverse matrix to solve the linear system:
A corporation has taxable income of $10,000,000.  The federal tax rate is 30%, the state tax rate is 8%, and the local tax rate is 4%.  The federal taxes are based on the income after the state and local taxes are deducted.  The state taxes are based on the income after the federal and local taxes are deducted.  The local taxes are based on the income after the federal  and state taxes are deducted.  What are the corporation's federal, state and local taxes?
Solution:
First identify the unknowns.  What is the question?  What are they asking you to solve for?  Since the problem asks for the corporation's taxes, let x = federal tax liability, y = state tax liability, and z = local tax liability.

Using the information, calculate the tax liabilities before deductions:
Federal Tax = 0.30($10,000,000) = $3,000,000
State Tax  = 0.08($10,000,000) = $800,000
Local Tax = 0.04($10,000,000) = $400,000

Allowing for the deductions:
x = $3,000,000 - 0.3y - 0.3z              x + 0.3y + 0.3z = $3,000,000
y = $800,000 - 0.08x - 0.08z         0.08x + y + 0.08z = $800,000
z = $400,000 - 0.04x - 0.04y         0.04x + 0.04y + z = $400,000

Set up matrixes A, X, B

Use the calculator to find .  

Store the inverse matrix in matrix D.
Use matrix multiplication to solve X = (B) .  

The federal tax liability is $2,752,202.09, the State tax liability is $558,417.82 and the local tax liability is $267,575.20.
Testing the solution,
x + 0.3y + 0.3z = $3,000,000     $2,752,202.09 + 0.3($558,417.82) + 0.3($267,575.20) = $3,000,000
0.08x + y + 0.08z = $800,000    0.08( $2,752,202.09) + $558,417.82 + 0.08($267,575.20) = $800,000
0.04x + 0.04y + z = $400,000    0.04( $2,752,202.09) + 0.04($558,417.82) + $267,575.20 = $400,000

The linear system is consistent and independent (only one solution).

Why Use Inverse Matrices to Solve Linear Systems?
Once you know the inverse matrix, it can be used to solve similar problems where the coefficients are the same, but the answers are 
different.  Consider another company in the same tax bracket at the corporation in Example 5 but their taxable income is $12,500,000.

Example 6:
Use the inverse matrix from Example 5 to solve the linear system:
A corporation has taxable income of $12,500,000.  The federal tax rate is 30%, the state tax rate is 8%, and the local tax rate is 4%.  The federal taxes are based on the income after the state and local taxes are deducted.  The state taxes are based on the income after the federal and local taxes are deducted.  The local taxes are based on the income after the federal  and state taxes are deducted.  What are the corporation's federal, state and local taxes?
Solution:
Using the information, calculate the tax liabilities before deductions:
Federal Tax = 0.30($12,500,000) = $3,750,000
State Tax  = 0.08($12,500,000) = $1,000,000
Local Tax = 0.04($12,500,000) = $500,000

The inverse matrix is stored in D.  

Use matrix multiplication to solve X = (B) .  Where .

 

 

The federal tax liability is $3,440,252.62, the State tax liability is $698,022.27 and the local tax liability is $334,469.00.
Testing the solution,
x + 0.3y + 0.3z = $3,000,000     $3,440,252.62 + 0.3($698,022.27) + 0.3($334,469.00) rounds to $3,750,000
0.08x + y + 0.08z = $800,000    0.08( $3,440,252.62) + $698,022.27 + 0.08($334,469.00) rounds to $1,000,000
0.04x + 0.04y + z = $400,000    0.04( $3,440,252.62) + 0.04($698,022.27) + $334,469.00 rounds to  $500,000

The linear system is consistent and independent (only one solution).

Thus the same inverse matrix can be used find the tax liabilities for all corporations in the same tax bracket as the corporation
in Example 5.

ASSIGNMENT:
Try solving various types of problems from Section 2-6:  1 through 49 odd

  Lesson 4          Lesson 6 

 


Copyright © 2003 Charlotte and Joseph Sukta. All rights reserved.
To contact authors: jsukta@morainevalley.edu

The content, views, and opinions on this page are strictly those of the author.
The contents of this page have not been reviewed or approved by Moraine Valley Community College.