Transcript Lesson 6.4

Lesson 6.3

Three friends, Duane, Marsha, and Parker,
decide to take their younger siblings to the
movies. Before the movie, they buy some
snacks at the concession stand.
◦ Duane buys two candy bars, a small drink, and two
boxes of chocolate-covered peanuts for a total of
$11.85.
◦ Marsha spends $9.00 on a candy bar, two small
drinks, and one box of chocolate-covered peanuts.
◦ Parker spends $12.35 on two small drinks and
three boxes of chocolate-covered peanuts, but
doesn’t buy any candy bars.

If all the prices include tax, what is the price
of each item?




Let c represent the price of a candy bar in
dollars
let d represent the price of a small drink in
dollars
let p represent the price of a box of
chocolate-covered peanuts in dollars.
This system represents the three friends’
purchases:

To use matrices to solve this system first,
translate these equations into a matrix
equation in the form [A][X]=[B]
2 1 2
c 


 
 A  1 2 1  X    c 
0 2 3
 p 
11.85


B    9.00 
12.35


Solving this equation [A][X]=[B] is similar to
solving the equation ax=b.
We will multiply both sides by the inverse of a
or a-1
 A  x   B 
ax  b
a
1

a xa
1
1xa
x  a1 b
b
1
b
 A
1
 A  x    A
 I   x    A
 x    A
1
1
1
B 
B 
B 
Where [I] in an identity matrix and [A]-1 in an
inverse matrix.

2 1 
Let’s find an identity matrix for 

4
3


2 1 
An identity matrix for 
matrix will be another 2 x

 4 3
2 matrix so that
Multiplying the two matrices yields



Because the two matrices are equal, their
entries must be equal. This yields:
By using substitution and elimination you can
find that a = 1, b = 0, c = 0, and d = 1.
Therefore
 a b  1 0
c d   0 1

 


1 0 
The 2 x 2 identity matrix is 

0
1


2
4

1
0

1 1 0 2




3 0 1  4
0  2 1  2




1   4 3  4
1
3
1
3
Can you see why multiplying this matrix by any
2 x 2 matrix results in the same 2 x 2 matrix?
Identity Matrix [I]: An identity matrix [I] is a square
matrix that does not change another square matrix
when multiplied.
If [A] is a given square matrix then [I] is an identity
matrix if
 A  I    I   A   A
1 0
 I   

0
1


1 0 0


 I   0 1 0
0 0 1 
Inverse Matrix [A]-1 : If [A] is a square matrix then
[A]-1 is the inverse matrix of [A] if
 A  A
1
1
  A  A   I 
Where [I] is an identity matrix.
1 0
 I   

0
1


1 0 0


 I   0 1 0
0 0 1 


In this investigation you will learn ways to
find the inverse of a 2 x 2 matrix.
Use the definition of an inverse matrix to set
up a matrix equation. Use these matrices and
the 2 x 2 identity matrix for [I].
2 1  a b  1 0
 4 3   c d   0 1 


 


Use matrix multiplication to find the product
[A][A]-1 . Set that product equal to matrix [I].
2b  d  1 0
 2a  c
4a  3c 4b  3d   0 1 

 



Use the matrix equation from the previous
step to write equations that you can solve to
find values for a, b, c, and d.
Solve the systems to find the values in the
inverse matrix.
2a+ c=1, 2b +d=0, 4a+ 3c=0, 4b+3d =1;
a =1.5, b=0.5, c=2, d=1;
 A
1
 1.5 0.5



2.0
1.0



Use your calculator to find [A ]-1 . If this
answer does not match your answer to the
last step, check your work for mistakes.



Find the products of [A][A]-1 and [A]-1[A].
Do they both give you 1?
Is matrix multiplication always commutative?

Not every square matrix has an inverse. Try
to find the inverse of each of these matrices.
Make a conjecture about what types of 2x2
square matrices do not have inverses.
None of the matrices has an inverse. A 2 x 2 square
matrix does not have an inverse when one row is a
multiple of the other.

Can a nonsquare matrix have an inverse? Why
or why not?
No. The product of a matrix and its inverse
must be a square matrix because an
identity matrix is always square and has
the dimensions of the matrix and its
inverse.


Solve this system using an inverse matrix.
First, rewrite the second equation in standard
form.

The matrix equation for this system is

If this equations corresponds with
 A  X   Constant 
find [A]-1 .
The solution to the system is (2, 1). Substitute the values
into the original equations to check the solution.


Use an inverse matrix to solve the problem
posed at the beginning of the lesson.
What is the cost of each snack item?
2 1 2
1 2 1


0 2 3
1
11.85  c 
 9.00   d 

  
12.35  p