4-7 INVERSE MATRICES AND SYSTEMS (p. 210-215)
A linear system of equations can be written in matrix form.
- 3x  7y  15
 3 7   x   15 
is equivalent to 

    
2x - 6y  -4
 2  6  y    4
Notice the second equation has the form AX = B.
Distinguish the coefficient matrix, the variable matrix, and the constant matrix.
Example: Write the system of three equations as a matrix equation. Put all variables in
the same order on the left. Zeros must be written as coefficients of missing variables.
Identify the coefficient, variable, and constant matrix.
3x - 4y  5z  11

- 2x  7y  -6
- 5x  y - z  20

Do 1b on p. 210.
If you can find the inverse of the coefficient matrix, then you can use this inverse to solve
the system of equations either manually or on the TI. Remember, if AX = B, then
X = A 1 B.
Example: Solve the following system the long way using the inverse matrix.
a b 
1  d - b
Recall that if A = 
, then A -1 
. There is no inverse if

ad - bc - c a 
c d 
ad – bc = 0.
2x  3y  -1

x - y  12
Do 2b on p. 211.
When dealing with a system of three or more equations, it is best to use the TI to solve it.
Example: Solve the following system on the TI. Store the coefficient matrix as
matrix A and the constant matrix as matrix B.
7x  3y  2z  13

- 2x  y - 8z  26
x - 4y  10z  -13

Do 3b on p. 211.
You can use matrices of systems of equations in the business profession.
Example: The Volunteer linen shop has sheets and towels on sale for the upcoming
football bowl game. Amy buys 3 sheets and 5 towels for $137.50. Travis buys 4 sheets
and 2 towels for $118.00. Write the corresponding system and matrix equation. Find the
price of each item. Use the TI.
Let x = the price for a sheet
$22.50
Let y = the price for a towel
$14.00
Do 4 on p. 212.
When the coefficient matrix of a system has an inverse, the system has a unique solution.
When the coefficient matrix does not have an inverse, the system does not have a unique
solution; it has either no solution or an infinite number of solutions (and you can not tell
which applies by just using the TI).
Example: Write the coefficient matrix for each system. Find its determinant to see
whether the system has a unique solution or not.
12x  8y  -3
1. 
3x - 7y  50
4x - 2y  7
2. 
- 6x  3y  5
Do 5a,b on p. 213.
Homework: p. 213-215: 5,11,13,16,21,22,25,27,33,37,38,43,45,46,51,53,56
37. Let x = length
y = width
x  2y

2x  2y  840
Note: 4-8 Augmented Matrices and Systems (p. 217-223). This section can be
skipped.