A fruit grower raises two crops, apples and peaches. Each of...

A fruit grower raises two crops, apples and peaches. Each of these crops is shipped to three different outlets. The number of units of crop $i$ that are shipped to outlet $j$ is represented by $a_{i j}$ in the matrix $$A=\left[\begin{array}{lll} 125 & 100 & 75 \\ 100 & 175 & 125 \end{array}\right]$$ The profit per unit is represented by the matrix $B=\left[\begin{array}{ll}\$ 3.50 & \$ 6.00\end{array}\right]$ Find the product $B A$ and state what each entry of the product represents.

A fruit grower raises two crops, apples and peaches. Each of these crops is shipped to three different outlets. The number of units of crop $i$ that are shipped to outlet $j$ is represented by $a_{i j}$ in the matrix
$$A=\left[\begin{array}{lll}
125 & 100 & 75 \\
100 & 175 & 125
\end{array}\right]$$
The profit per unit is represented by the matrix
$B=\left[\begin{array}{ll}\$ 3.50 & \$ 6.00\end{array}\right]$
Find the product $B A$ and state what each entry of the product represents.

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Key Concepts

Matrix Multiplication

Matrix multiplication is a key linear algebra operation where elements from the rows of the first matrix are multiplied with corresponding elements from the columns of the second matrix and summed. In this context, multiplying the row vector of profits with the shipment matrix aggregates the total profit for each outlet by summing the contributions from each crop based on their shipment numbers and corresponding profit per unit.

Interpretation of Matrix Product Entries

Each entry of the resulting product matrix represents the total profit generated from a specific outlet. The computation involves summing the profit contributions of all crops shipped to that outlet, thus providing a consolidated profit figure for each outlet based on the number of units shipped and the profit per unit.

Matrix Representation in Applied Context

Matrices are used to organize and structure data in many real-world applications. In this problem, one matrix represents quantities (the number of units shipped from each crop to various outlets), while another represents the profit per unit for each crop. This structured format allows for systematic computation and clear association of data points, making it easier to perform overall calculations such as total profit.

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