The Bread Box, a small neighborhood bakery, sells four main...

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 $A$. The cost (in cents) for each ingredient when purchased in either large lots or small lots is given in matrix $B$. (a) Use matrix multiplication to find a matrix giving the cost per 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. FIGURE CANNOT COPY.

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 $A$.
The cost (in cents) for each ingredient when purchased in either large lots or small lots is given in matrix $B$.
(a) Use matrix multiplication to find a matrix giving the cost per 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.
FIGURE CANNOT COPY.

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra that combines two matrices to produce a new matrix. The entry in the resulting matrix is computed as the dot product of a row from the first matrix and a column from the second matrix. This operation is useful for combining data in structured ways, such as calculating total costs by balancing ingredient amounts with their unit costs.

Conforming Dimensions

For the product of two matrices to be defined, the number of columns in the first matrix must equal the number of rows in the second matrix. This concept of dimensional conformity is crucial when setting up matrix equations, such as when calculating the cost per item or the total ingredients required, to ensure that the mathematical operations are valid.

Matrix Representation of Systems

Matrices can be used to represent systems of linear equations or multiple related quantities compactly. In the context of a bakery calculating ingredient requirements and costs, matrices can encode the recipe for each item (ingredient amounts) and the cost per ingredient. This structured representation facilitates efficient computation and analysis of the entire system.

Application to Cost and Resource Calculations

By using matrix multiplication, one can combine a matrix that represents ingredient usage with another that provides ingredient costs to obtain total cost figures. Similarly, representing orders as a matrix and multiplying by a recipe matrix yields the total required quantities of each ingredient. This illustrates the powerful application of matrices in practical decision-making and resource management.

Transcript

Please give Ace some feedback