Pauline, Patricia and Polly are pie bakers at their bakery, Perfect Pies. They bake four different pie flavors: cherry, blueberry, apple and pecan.
This table shows the number of the different pie flavors each woman made.
Baker Cherry Blueberry Apple Pecan
Pauline 3 4 1 1
Patricia 6 6 4 6
Polly 2 2 5 4

This table shows the cost of ingredients for one of each of the pies.
Type of Pie Cost of ingredients
Cherry $3.00
Blueberry $5.00
Apple $2.00
Pecan $6.00

This table shows the price at which these pies are sold to 2 different stores.
Type of Pie Samuel’s Sweets Sugary Sarah
Cherry $8.00 $8.00
Blueberry $10.00 $9.00
Apple $7.00 $8.00
Pecan $12.00 $13.00
1. Using matrix multiplication, create the matrix that represents the cost of ingredients used by each person to make their pies. Create matrix P to represents the person, and matrix I to represent cost of ingredients. Then multiply to determine matrix .

2. Using , what is the cost of ingredients for the pies Polly made?

3. Using , determine whether the cost of ingredients for the pies Patricia baked is more or less than half the cost of ingredients for all pies baked by the three bakers.

4. Using matrix multiplication, create the matrix that represents the income of each baker from each of the 2 stores at which Perfect Pies are sold.

5. Using , which baker yielded the greatest income? What is this income?

6. Using , what income did Polly realize from Sugary Sarah’s?

\[ P \cdot I = \begin{pmatrix} 3 & 4 & 1 & 1 \\ 6 & 6 & 4 & 6 \\ 2 & 2 & 5 & 4 \end{pmatrix} \cdot \begin{pmatrix} 3.00 \\ 5.00 \\ 2.00 \\ 6.00 \end{pmatrix} = \begin{pmatrix} 3(3) + 4(5) + 1(2) + 1(6) \\ 6(3) + 6(5) + 4(2) + 6(6) \\ 2(3) + 2(5) + 5(2) + 4(6) \end{pmatrix} = \begin{pmatrix} 47 \\ 102 \\ 64 \end{pmatrix} \]

2. The cost of ingredients for the pies Polly made is $64.

3. Let's calculate the total cost of ingredients for all pies baked by the three bakers:

\[ Total\ Cost = 47 + 102 + 64 = 213 \]

Half of the total cost of ingredients is $106.50. Now, let's compare this to the cost of ingredients for the pies Patricia baked:

\[ \text{Cost for Patricia} = 47 + 102 + 64 = 213 \]

Since $213 is equal to half of the total cost, the cost of ingredients for the pies Patricia baked is not more or less than half the cost of ingredients for all pies baked by the three bakers.

4. Let's create the matrix representing the income of each baker from each store:

Matrix S (representing Samuel's Sweets):

\[ S = \begin{pmatrix} 8 & 8 \\ 10 & 9 \\ 7 & 8 \\ 12 & 13 \end{pmatrix} \]

Matrix SS (representing Sugary Sarah):

\[ SS = \begin{pmatrix} 8 & 8 \\ 10 & 9 \\ 7 & 8 \\ 12 & 13 \end{pmatrix} \]

5. Now, let's find out which baker yielded the greatest income:

\[ S \cdot P^T = \begin{pmatrix} 8 & 8 \\ 10 & 9 \\ 7 & 8 \\ 12 & 13 \end{pmatrix} \cdot \begin{pmatrix} 3 & 6 & 2 \\ 4 & 6 & 2 \\ 1 & 4 & 5 \\ 1 & 6 & 4 \end{pmatrix} = \begin{pmatrix} 47 & 92 & 39 \\ 62 & 126 & 55 \\ 56 & 116 & 63 \\ 88 & 168 & 82 \end{pmatrix} \]

The greatest income is yielded by the third baker (Polly), which is $116 from Samuel's Sweets.

6. Now, let's find out the income Polly realized from Sugary Sarah's:

\[ SS \cdot P^T = \begin{pmatrix} 8 & 8 \\ 10 & 9 \\ 7 & 8 \\ 12 & 13 \end{pmatrix} \cdot \begin{pmatrix} 3 & 6 & 2 \\ 4 & 6 & 2 \\ 1 & 4 & 5 \\ 1 & 6 & 4 \end{pmatrix} = \begin{pmatrix} 47 & 92 & 39 \\ 62 & 126 & 55 \\ 56 & 116 & 63 \\ 88 & 168 & 82 \end{pmatrix} \]

The income Polly realized from Sugary Sarah's is $63.