Q7(a):

A manufacturer produces three products $x, y, z$ which he sells in two markets. Annual sales are indicated below: 

(a) If unit sale prices of $x, y$ and $z$ are ` 2.50, ` 1.50 and ` 1.00, respectively, find the total revenue in each market with the help of matrix algebra.

Step 1: Matrix Representation of Quantity and Price

To determine the total revenue using matrix algebra, we define the quantity sold in the two markets as a matrix $A$ and the unit sale prices as a row matrix $P$.

Let the quantity matrix $A$ be a $2 \times 3$ matrix representing Markets (rows) and Products (columns):

$A = \begin{bmatrix} 10000 & 2000 & 18000 \\ 6000 & 20000 & 8000 \end{bmatrix}$

Let the price matrix $P$ be a $3 \times 1$ column matrix representing the unit price per product $x, y,$ and $z$:

$P = \begin{bmatrix} 2.50 \\ 1.50 \\ 1.00 \end{bmatrix}$

Step 2: Revenue Calculation via Matrix Multiplication

The total revenue $R$ for each market is obtained by the matrix product $R = A \times P$. [Per the definition of matrix multiplication, where an $(m \times n)$ matrix multiplied by an $(n \times p)$ matrix results in an $(m \times p)$ matrix].

$R = \begin{bmatrix} 10000 & 2000 & 18000 \\ 6000 & 20000 & 8000 \end{bmatrix} \begin{bmatrix} 2.50 \\ 1.50 \\ 1.00 \end{bmatrix}$

Step 3: Execution of Algebraic Operations

Perform row-by-column multiplication:

For Market 1:

$R_{1} = (10000 \times 2.50) + (2000 \times 1.50) + (18000 \times 1.00)$

$R_{1} = 25000 + 3000 + 18000 = 46000$

For Market 2:

$R_{2} = (6000 \times 2.50) + (20000 \times 1.50) + (8000 \times 1.00)$

$R_{2} = 15000 + 30000 + 8000 = 53000$

Step 4: Visualizing the Matrix Revenue Mapping

Summary of Revenue

Final Solution: The total revenue for Market 1 is ₹46,000 and for Market 2 is ₹53,000.