Profit A discount furniture store sells four types of 5...

Profit A discount furniture store sells four types of 5 -piece bedroom sets. The price charged for a bedroom set of type $j$ is represented by $a_{1 j}$ in the matrix $$ A=\left[\begin{array}{llll}{\$ 398} & {\$ 598} & {\$ 798} & {\$ 998}\end{array}\right] $$ The number of sets of type $j$ sold in one period is represented by $b_{1 j}$ in the matrix $$ B=\left[\begin{array}{llll}{35} & {25} & {20} & {10}\end{array}\right] $$ The cost to the furniture store for a bedroom set of type $j$ is given by $c_{1 /}$ in the matrix $$ C=\left[\begin{array}{llll}{\mathrm{S} 199} & {\$ 268} & {\$ 500} & {\$ 670}\end{array}\right] $$ (a) Write a matrix product that gives the total revenue made from the sale of the bedroom sets in the one period. (b) Write an expression using matrices that gives the profit produced by the sale of the bedroom sets in the one period.

Profit A discount furniture store sells four types of 5 -piece bedroom sets. The price charged for a bedroom set of type $j$ is represented by $a_{1 j}$ in the matrix
$$
A=\left[\begin{array}{llll}{\$ 398} & {\$ 598} & {\$ 798} & {\$ 998}\end{array}\right]
$$
The number of sets of type $j$ sold in one period is represented by $b_{1 j}$ in the matrix
$$
B=\left[\begin{array}{llll}{35} & {25} & {20} & {10}\end{array}\right]
$$
The cost to the furniture store for a bedroom set of type $j$ is given by $c_{1 /}$ in the matrix
$$
C=\left[\begin{array}{llll}{\mathrm{S} 199} & {\$ 268} & {\$ 500} & {\$ 670}\end{array}\right]
$$
(a) Write a matrix product that gives the total revenue made from the sale of the bedroom sets in the one period.
(b) Write an expression using matrices that gives the profit produced by the sale of the bedroom sets in the one period.

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Key Concepts

Matrix Expression for Profit Calculation

Using matrix expressions to calculate profit involves formulating the revenue and cost computations as matrix products and then subtracting the total cost from the total revenue. This technique leverages both matrix multiplication and element-wise subtraction to provide a systematic and scalable way to compute profit when dealing with multiple products and their associated pricing, quantities sold, and cost data.

Matrix Multiplication

Matrix multiplication is a foundational operation in linear algebra where two matrices are combined to produce a new matrix. This process involves taking the dot product of rows from the first matrix with columns from the second matrix. It is essential for translating multiple business quantities into a single summary value, such as calculating total revenue by multiplying price and quantity vectors or arrays.

Dot Product

The dot product is a specific type of matrix multiplication for two vectors. It involves multiplying corresponding entries of two sequences of numbers and then summing the results. In a business context, the dot product can be used to compute the total revenue by multiplying individual prices by the corresponding quantities sold and summing those products.

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