Example 11: Two farmers' rice sales

Basics you need

Part (i): combined sales, $A+B$

1. Read the two matrices and check that the sum is defined

   Both $A$ and $B$ have 2 rows (the farmers) and 3 columns (the varieties), so both have order $2\times3$⁠. The same order means $A+B$ is defined, and it is again a $2\times3$ matrix with the same row and column labels.

   Entry: farmer, variety	Sept. $a_{ij}$	Oct. $b_{ij}$
   $(1,1)$⁠: Ramkishan, Basmati	10,000	5000
   $(1,2)$⁠: Ramkishan, Permal	20,000	10,000
   $(1,3)$⁠: Ramkishan, Naura	30,000	6000
   $(2,1)$⁠: Gurcharan Singh, Basmati	50,000	20,000
   $(2,2)$⁠: Gurcharan Singh, Permal	30,000	10,000
   $(2,3)$⁠: Gurcharan Singh, Naura	10,000	10,000

   "Combined sales for each farmer in each variety" means: for every (farmer, variety) pair, add the September amount to the October amount. Because both matrices put the same pair in the same position, adding position by position is exactly that. If the two tables had their columns in different orders, entry-wise addition would mix up varieties and give nonsense.
   Addition is defined only for matrices of the same order. Basics: adding only like with like

2. Add entry by entry

   Write $A+B=[c_{ij}]$ with $c_{ij}=a_{ij}+b_{ij}$⁠:

   $$\begin{aligned} c_{11} &= 10{,}000 + 5000 = 15{,}000\\ c_{12} &= 20{,}000 + 10{,}000 = 30{,}000\\ c_{13} &= 30{,}000 + 6000 = 36{,}000\\ c_{21} &= 50{,}000 + 20{,}000 = 70{,}000\\ c_{22} &= 30{,}000 + 10{,}000 = 40{,}000\\ c_{23} &= 10{,}000 + 10{,}000 = 20{,}000 \end{aligned}$$

   Assembling the six entries in their positions:

   $$A+B = \begin{bmatrix} 15{,}000 & 30{,}000 & 36{,}000 \\ 70{,}000 & 40{,}000 & 20{,}000 \end{bmatrix}$$
   Each entry of the sum uses only the two entries in the same position, so there are exactly six small additions, one per farmer and variety. Nothing from another row or column is involved.
   Definition of the sum: if $A=[a_{ij}]$ and $B=[b_{ij}]$ are both $m\times n$⁠, then $A+B=[a_{ij}+b_{ij}]$⁠, Section 3.4.1. Lesson: addition and scalar multiplication

3. Say what the sum means

   Over September and October together, Ramkishan sold ₹15,000 of Basmati, ₹30,000 of Permal and ₹36,000 of Naura, and Gurcharan Singh sold ₹70,000 of Basmati, ₹40,000 of Permal and ₹20,000 of Naura.

   The labels travel with the positions: row 1 is still Ramkishan and column 3 is still Naura. Reading the answer back in words, with the unit, is how we confirm the matrix answers the question that was asked.

Part (ii): the decrease from September to October, $A-B$

1. Decide which difference to take

   A decrease "from September to October" is the September amount minus the October amount, so we want $A-B$⁠, not $B-A$⁠. Both are $2\times3$⁠, so the difference is defined and is $2\times3$⁠.

   If sales fall from ₹10,000 to ₹5000, the decrease is $10{,}000-5000=5000$⁠, a positive number. Subtracting the later amount from the earlier one gives the decrease directly. $B-A$ would give October minus September, which is the negative of the decrease. (The textbook's solution heads this part "Change in sales from September to October" and computes $A-B$⁠, the same matrix as here.)
   The difference is $A-B=[a_{ij}-b_{ij}]$⁠, which is the same as $A+(-1)B$⁠, Section 3.4.2. Basics: adding only like with like

2. Subtract entry by entry

   Write $A-B=[d_{ij}]$ with $d_{ij}=a_{ij}-b_{ij}$⁠:

   $$\begin{aligned} d_{11} &= 10{,}000 - 5000 = 5000\\ d_{12} &= 20{,}000 - 10{,}000 = 10{,}000\\ d_{13} &= 30{,}000 - 6000 = 24{,}000\\ d_{21} &= 50{,}000 - 20{,}000 = 30{,}000\\ d_{22} &= 30{,}000 - 10{,}000 = 20{,}000\\ d_{23} &= 10{,}000 - 10{,}000 = 0 \end{aligned}$$ $$A-B = \begin{bmatrix} 5000 & 10{,}000 & 24{,}000 \\ 30{,}000 & 20{,}000 & 0 \end{bmatrix}$$
   Again each entry depends only on the two entries in the same position. Every result is $\ge 0$⁠, so in every case the October sale is no larger than the September sale: the word "decrease" fits every entry, and nothing went up.
   Order matters in subtraction of numbers ($5-3\ne3-5$⁠), and therefore in subtraction of matrices: $B-A=-(A-B)$⁠.

3. Say what the difference means

   From September to October, Ramkishan's sales fell by ₹5000 (Basmati), ₹10,000 (Permal) and ₹24,000 (Naura); Gurcharan Singh's fell by ₹30,000 (Basmati) and ₹20,000 (Permal), and his Naura sales did not change (decrease ₹0).

   An entry 0 in a difference matrix means the two months were equal for that farmer and variety: $10{,}000$ in both months for Gurcharan Singh's Naura.

Part (iii): 2% profit on the October sales, $0.02B$

1. Turn "2% of each October sale" into a scalar multiple

   "Gross sales" means the full amount received from sales. A profit of 2% on gross sales is $\tfrac{2}{100}=0.02$ of each sale amount. We want this for every farmer and variety in October, that is for every entry of $B$⁠, so the profit matrix is

   $$\frac{2}{100}\times B = 0.02\,B.$$

   Multiplying by a number does not change the order, so $0.02B$ is $2\times3$ with the same labels.

   Taking 2% of every entry is the same as multiplying every entry by $0.02$⁠, and that is exactly what a scalar multiple does. Only October is asked for, so we scale $B$⁠, not $A$ or $A+B$⁠.
   Scalar multiple: $kB=[k\,b_{ij}]$⁠, Section 3.4.2. Lesson: addition and scalar multiplication

2. Multiply every entry by 0.02

   Write $0.02B=[p_{ij}]$ with $p_{ij}=0.02\,b_{ij}$⁠. Multiplying by $0.02$ is the same as multiplying by 2 and dividing by 100:

   $$\begin{aligned} p_{11} &= 0.02\times 5000 = 100\\ p_{12} &= 0.02\times 10{,}000 = 200\\ p_{13} &= 0.02\times 6000 = 120\\ p_{21} &= 0.02\times 20{,}000 = 400\\ p_{22} &= 0.02\times 10{,}000 = 200\\ p_{23} &= 0.02\times 10{,}000 = 200 \end{aligned}$$ $$0.02\,B = \begin{bmatrix} 100 & 200 & 120 \\ 400 & 200 & 200 \end{bmatrix}$$
   For example $0.02\times5000$ is $\tfrac{2\times5000}{100}$⁠, that is $\tfrac{10{,}000}{100}=100$⁠. Doing the division by 100 last keeps the arithmetic in whole numbers.
   $2\% = \tfrac{2}{100} = 0.02$ (Basic 4).

3. Say what the profit matrix means

   In October, Ramkishan receives ₹100, ₹200 and ₹120 as profit on Basmati, Permal and Naura, and Gurcharan Singh receives ₹400, ₹200 and ₹200.

   Adding along a row gives each farmer's total October profit: $100+200+120=420$ rupees for Ramkishan and $400+200+200=800$ rupees for Gurcharan Singh. The question asks for the profit for each variety, so the matrix itself is the answer; the row totals are a useful extra.

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