Solve each problem. Purchasing costs The Bread Box, a small...

Solve each problem. Purchasing costs The Bread Box, a small neighborhood bakery, sells four main items: sweet rolls, bread, cakes, and pies. The amount of each ingredient (in cups, except for eggs) required for these items is given by matrix $A$ Eggs lour Sugar Shortening Milk $\left.\begin{array}{l|ccccc}\text { Rolls (doz) } & 1 & 4 & \frac{1}{4} & \frac{1}{4} & 1 \\ \text { Bread (loaf) } & 0 & 3 & 0 & \frac{1}{4} & 0 \\ \text { Cake } & 4 & 3 & 2 & 1 & 1 \\ \text { Pie (crust) } & 0 & 1 & 0 & \frac{1}{3} & 0\end{array}\right]=A$ The cost (in cents) for each ingredient when purchased in large lots or small lots is given by matrix $B$ Large Lot Small Lot $\left.\begin{array}{l|rr}\text { Eggs } & 5 & 5 \\ \text { Flour } & 8 & 10 \\ \text { Sugar } & 10 & 12 \\ \text { Shortening } & 12 & 15 \\ \text { Milk } & 5 & 6\end{array}\right]=B$ (a) Use matrix multiplication to find a matrix giving the comparative cost per bakery item for the two purchase options. (b) Suppose a day's orders consist of 20 dozen sweet rolls, 200 loaves of bread, 50 cakes, and 60 pies. Write the orders as a $1 \times 4$ matrix, and, using matrix multiplication, write as a matrix the amount of each ingredient needed to fill the day's orders. (c) Use matrix multiplication to find a matrix giving the costs under the two purchase options to fill the day's orders.

Solve each problem.
Purchasing costs The Bread Box, a small neighborhood bakery, sells four main items: sweet rolls, bread, cakes, and pies. The amount of each ingredient (in cups, except for eggs) required for these items is given by matrix $A$
Eggs lour Sugar Shortening Milk $\left.\begin{array}{l|ccccc}\text { Rolls (doz) } & 1 & 4 & \frac{1}{4} & \frac{1}{4} & 1 \\ \text { Bread (loaf) } & 0 & 3 & 0 & \frac{1}{4} & 0 \\ \text { Cake } & 4 & 3 & 2 & 1 & 1 \\ \text { Pie (crust) } & 0 & 1 & 0 & \frac{1}{3} & 0\end{array}\right]=A$

The cost (in cents) for each ingredient when purchased in large lots or small lots is given by matrix $B$

Large Lot Small Lot
$\left.\begin{array}{l|rr}\text { Eggs } & 5 & 5 \\ \text { Flour } & 8 & 10 \\ \text { Sugar } & 10 & 12 \\ \text { Shortening } & 12 & 15 \\ \text { Milk } & 5 & 6\end{array}\right]=B$
(a) Use matrix multiplication to find a matrix giving the comparative cost per bakery item for the two purchase options.
(b) Suppose a day's orders consist of 20 dozen sweet rolls, 200 loaves of bread, 50 cakes, and 60 pies. Write the orders as a $1 \times 4$ matrix, and, using matrix multiplication, write as a matrix the amount of each ingredient needed to fill the day's orders.
(c) Use matrix multiplication to find a matrix giving the costs under the two purchase options to fill the day's orders.

Key Concepts

Matrix Multiplication

Matrix multiplication is a core operation in linear algebra where each entry of the resulting matrix is obtained by taking the dot product of rows from the first matrix with columns from the second matrix. This operation is essential for combining different sets of data and transformations, and it enables the concise representation and solution of complex problems through sequential applications of linear mappings.

Representing Systems with Matrices

Using matrices to represent systems of linear relationships allows for multiple variables and constraints to be compactly formulated and solved simultaneously. This approach is particularly useful in contexts where there are several interrelated components, as each matrix can encapsulate different aspects of the system, making the analysis and computation straightforward via matrix operations.

Application of Matrix Techniques in Cost and Quantity Analysis

By organizing unit costs, resource requirements, or conversion factors into matrices, problems involving aggregate computations—such as calculating total costs or total inputs—can be efficiently addressed using matrix multiplication. This technique not only simplifies the arithmetic but also provides a scalable method for handling complex, multi-variable scenarios in various fields such as economics, engineering, and supply chain management.

Matrix Dimension Compatibility

A fundamental principle in performing matrix multiplication is that the number of columns in the first matrix must equal the number of rows in the second matrix. This compatibility condition ensures that the multiplication is well-defined and that each element in the resulting matrix correctly reflects the interactions between the corresponding elements of the original matrices.

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