Download 2-5 Determinants and Multiplicative Inverses of Matrices

Advanced Mathematical Concepts
Chapter 2
Lesson 2-5
Example 1
Find the value of
3 5
2 9
.
3 5
= 3(9) - 2(5) or 17
2 9
Example 2
2 -3 -5
Find the value of 1
5
2 3 5
2 =2
1
2
5
3 1
2
2
2.
3 -1
2
3 1
- (-3)
1
2
5 1
+ (-5)
= 2(-8) + 3(-11) – 5(-7)
= -14
Example 3
4 2
Find the inverse of the matrix 
.
3 2
4 2
First, find the determinant of 
.
3 2
4 2
3 2
= 4(2) - 3(2) or 2
 1 1
1  2 2 
.
The inverse is 
or  3

2  3 4 
2
 2

1 2
5 3
Advanced Mathematical Concepts
Chapter 2
Example 4
Solve the system of equations by using matrix equations.
3x + 2y = 3
2x – 4y = 2
Write the system as a matrix equation.
 3 2  x 
 3
 2 4    y    2 

  
 
To solve the matrix equation, first find the inverse of the coefficient matrix.
 4 2
1
1  4 2  3
  



3 2  2 3
16  2 3  2
2 4
Now multiply each side of the matrix equation by the inverse and solve.
1  4 2   3 2   x 
1  4 2   3
 
 
   =  



3  2 4   y 
3  2 
16  2
16  2
 x
 1
 y  = 0 
 
 
The solution is (1, 0).
Example 5
BANKING A teller at Security Bank received a deposit from a local retailer containing only
twenty-dollar bills and fifty-dollar bills. He received a total of 70 bills, and the amount of the
deposit was $3200. How many bills of each value were deposited?
First, let x represent the number of twenty-dollar bills and let y represent the number of
fifty-dollar bills. So, x + y = 70 since a total of 70 bills were deposited.
Write an equation in standard form that represents the total amount deposited.
20x + 50y = 3,200
Now solve the system of equations x + y = 70 and 20x + 50y = 3200. Write the system as a matrix
equation and solve.
x + y = 70
 1 1  x 
 70 
20x + 50y = 3200
 20 50    y  = 3200 

  


Multiply each side of the
equation by the inverse of the
coefficient matrix.
1  50 1  1 1  x 
1  50 1  70 
 
   =




30  20 1  20 50   y 
30  20 1 3200 
 x
10 
 y  = 60 
 
 
The solution is (10, 60). The deposit contained 10 twenty-dollar bills and 60 fifty-dollar bills.