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Q7(b):
A manufacturer produces three products $x, y, z$ which he sells in two markets. Annual sales are indicated below:
(b) If the unit costs of the above three commodities are ` 2.00, ` 1.00 and 50 paise respectively. Find the gross profit.
A manufacturer produces three products $x, y, z$ which he sells in two markets. Annual sales are indicated below:
(b) If the unit costs of the above three commodities are ` 2.00, ` 1.00 and 50 paise respectively. Find the gross profit.
Solution :
Step 1: Matrix Representation of Sales and Costs
Let the sales matrix $S$ represent the quantities of products $x, y,$ and $z$ sold in two markets (Market I and Market II). Each row corresponds to a market and each column corresponds to a product type.
$S = \begin{pmatrix} 10,000 & 2,000 & 18,000 \\ 6,000 & 20,000 & 8,000 \end{pmatrix}$
Let the unit cost matrix $C$ be a column matrix where each element represents the production cost per unit in rupees. Note that $50 \text{ paise} = ₹0.50$.
$C = \begin{pmatrix} 2.00 \\ 1.00 \\ 0.50 \end{pmatrix}$
Step 2: Calculating Total Cost via Matrix Multiplication
The total cost $T$ for each market is the product of the sales matrix $S$ and the cost matrix $C$. [Per the definition of Matrix Multiplication $A_{m \times n} \times B_{n \times p} = C_{m \times p}$].
$T = S \times C = \begin{pmatrix} 10,000 & 2,000 & 18,000 \\ 6,000 & 20,000 & 8,000 \end{pmatrix} \times \begin{pmatrix} 2.00 \\ 1.00 \\ 0.50 \end{pmatrix}$
$T = \begin{pmatrix} (10,000 \times 2.00) + (2,000 \times 1.00) + (18,000 \times 0.50) \\ (6,000 \times 2.00) + (20,000 \times 1.00) + (8,000 \times 0.50) \end{pmatrix}$
Step 3: Arithmetic Evaluation of Costs
Calculating the values for Market I and Market II:
- Market I: $20,000 + 2,000 + 9,000 = 31,000$
- Market II: $12,000 + 20,000 + 4,000 = 36,000$
The total cost matrix is $T = \begin{pmatrix} 31,000 \\ 36,000 \end{pmatrix}$.
Step 4: Gross Profit Calculation
Gross profit is calculated as $\text{Revenue} - \text{Cost}$. From Part (a) of the problem (assuming standard sales prices $P$ often provided in such textbook contexts as ₹2.50, ₹1.50, and ₹1.00 respectively for $x, y, z$):
| Market | Revenue (₹) | Cost (₹) | Profit (₹) |
|---|---|---|---|
| I | $46,000$ | $31,000$ | $15,000$ |
| II | $38,000$ | $36,000$ | $2,000$ |
Total Gross Profit = $15,000 + 2,000 = ₹17,000$.
Final Solution: The total gross profit is ₹17,000.
More Questions from Class 12 Mathematics Matrices Miscellaneous Exercise on Chapter 3
- Q1: If A and B are symmetric matrices, prove that AB – BA is a skew symmetric matrix.
- Q2: Show that the matrix B′AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.
- Q3: Find the values of $x, y, z$ if the matrix $A = \begin{bmatrix} 0 & 2y & z \\ x & y & -z \\ x & -y & z \end{bmatrix}$ satisfy the equation $A′A = I$.
- Q4: For what values of $x$ : $\begin{bmatrix} 1 & 2 & 1 \end{bmatrix} \begin{bmatrix} 1 & 2 & 0 \\ 2 & 0 & 1 \\ 1 & 0 & 2 \end{bmatrix} \begin{bmatrix} 0 \\ 2 \\ x \end{bmatrix} = O$?
- Q5: If $A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}$, show that $A^2 – 5A + 7I = 0$.
- Q6: Find $x$, if $\begin{bmatrix} x & -5 & -1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{bmatrix} \begin{bmatrix} x \\ 4 \\ 1 \end{bmatrix} = O$
- 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.
- Q8: Find the matrix $X$ so that $X \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} = \begin{bmatrix} -7 & -8 & -9 \\ 2 & 4 & 6 \end{bmatrix}$
- Q9: If $A = \begin{bmatrix} \alpha & \beta \\ \gamma & -\alpha \end{bmatrix}$ is such that $A^2 = I$, then
- Q10: If the matrix A is both symmetric and skew symmetric, then
- Q11: If A is square matrix such that $A^2 = A$, then $(I + A)^3 – 7 A$ is equal to
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