Two farmers Ramkishan and Gurcharan Singh cultivates only three varieties of rice namely Basmati, Permal and Naura. The sale (in Rupees) of these varieties of rice by both the farmers in the month of September and October are given by the following matrices $A$ and $B$. September

Question

Two farmers Ramkishan and Gurcharan Singh cultivates only three varieties of rice namely Basmati, Permal and Naura. The sale (in Rupees) of these varieties of rice by both the farmers in the month of September and October are given by the following matrices $A$ and $B$. September Sales (in Rupees), columns Basmati, Permal, Naura, rows Ramkishan, Gurcharan Singh: $A = \begin{bmatrix} 10000 & 20000 & 30000 \ 50000 & 30000 & 10000 \end{bmatrix}$. October Sales (in Rupees): $B = \begin{bmatrix} 5000 & 10000 & 6000 \ 20000 & 10000 & 10000 \end{bmatrix}$. (i) Find the combined sales in September and October for each farmer in each variety. (ii) Find the decrease in sales from September to October. (iii) If both farmers receive $2\%$ profit on gross sales, compute the profit for each farmer and for each variety sold in October.

Accepted Answer

> [!TLDR]
> Matrix addition and subtraction give combined sales and decrease in sales directly. For the profit calculation, multiply each October entry by $0.02$ (i.e., $2\%$). The combined sales matrix is $A+B$, the decrease is $A-B$, and the October profit matrix is $0.02 \times B$.

## Why matrix addition works here

Each matrix is a compact table: rows are farmers (Ramkishan, Gurcharan Singh), columns are rice varieties (Basmati, Permal, Naura). The entry at row $i$, column $j$ is the sale in Rupees for that farmer and that variety in the given month.

When we want the **total** sales over two months, we add the corresponding entries — because each entry is a separate quantity (a specific farmer’s sale of a specific variety). There’s no mixing across farmers or varieties. That’s exactly what matrix addition does: add element by element.

Similarly, the **decrease** from September to October is the difference: September minus October, entry by entry.

For the profit, $2\%$ of a sale means multiplying the sale amount by $0.02$. Since profit is computed per farmer per variety, we just scale every entry of the October matrix by $0.02$.

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## Step-by-step solution

**1. Combined sales (September + October)**

Add $A$ and $B$ entry-wise:

$$
A + B = \begin{bmatrix}
10000 + 5000 & 20000 + 10000 & 30000 + 6000 \[4pt]
50000 + 20000 & 30000 + 10000 & 10000 + 10000
\end{bmatrix}
$$

$$
= \begin{bmatrix}
15000 & 30000 & 36000 \[4pt]
70000 & 40000 & 20000
\end{bmatrix}
$$

So Ramkishan’s total sales: Basmati ₹15,000, Permal ₹30,000, Naura ₹36,000.  
Gurcharan Singh’s total: Basmati ₹70,000, Permal ₹40,000, Naura ₹20,000.

> [!TIP]
> Always check dimensions: both $A$ and $B$ are $2 \times 3$, so their sum is also $2 \times 3$. If dimensions didn’t match, addition would be undefined.

**2. Decrease in sales (September – October)**

Subtract $B$ from $A$ entry-wise:

$$
A - B = \begin{bmatrix}
10000 - 5000 & 20000 - 10000 & 30000 - 6000 \[4pt]
50000 - 20000 & 30000 - 1000

Time	Apples	Bananas
Morning	10	5
Afternoon	8	12

Concept understanding — Matrix Addition

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Method: Using matrix add / subtract / scalar-multiply on data tables

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Why matrix addition works here

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