Theory
Rupa's one-glance report
Rupa (from the multiplication topic) now wants a monthly report for her Campus and Station counters: combined sales, growth, revenue per counter, and what happens if prices rise.
Her competitor scribbles on four pages of loose arithmetic. Rupa writes two grids and four short operations.
You already run your life on this idea: every Excel sheet you have ever used is a matrix wearing office clothes.
Theory
Rows are things, columns are facts
Every business grid follows one convention: rows are the things you compare (branches, counters, months) and columns are the facts about them (items sold, costs, marks). Fix that shape once and keep it. The moment two reports disagree about what a row means, every calculation after that is quietly lying.
At a glance
Which operation answers which question
| Business question | Matrix move | With Rupa's data |
|---|---|---|
| Two months combined? | Addition: Jan + Feb | Total sales table |
| How much did we grow? | Subtraction: Feb - Jan | Growth per counter, per item |
| Revenue per counter? | Multiplication: S·P | Sales grid × price column |
| All prices double? | Scalar multiple: 2P | One number scales every cell |
Theory
Worked example: February revenue
February sales S and prices P (samosa ₹9, tea ₹7):
S = ⎡ 12 9 ⎤ P = ⎡ 9 ⎤
⎣ 10 14 ⎦ ⎣ 7 ⎦
Inner numbers match (2 = 2), answer is 2×1:
Campus: 12·9 + 9·7 = 108 + 63 = 171
Station: 10·9 + 14·7 = 90 + 98 = 188
S·P = ⎡ 171 ⎤
⎣ 188 ⎦
Read it like a manager: the Station counter earns more, mostly on tea volume.
Think first
Your turn: read the growth matrix
January sales were rows (10 6) and (8 12). February: rows (12 9) and (10 14).
Compute Feb - Jan, then answer like an analyst: which single cell grew the most, and what does it mean in words?
Show the answer
Feb - Jan = ⎡ 2 3 ⎤
⎣ 2 2 ⎦
The biggest cell is 3, at row 1, column 2: the Campus counter sold 3 (hundred) more teas than in January. A matrix answer is not finished until you can say it in a sentence a shopkeeper would nod at.
Quiz
A chain has a 3×4 sales matrix (3 shops, 4 items). What must it be multiplied by to get revenue per shop?
- A 4×1 price column
- A 1×3 price row
- Another 3×4 matrix
- A 3×3 matrix
Show the answer
A 4×1 price column
Revenue needs one price per item, and the inner numbers must match: (3×4)·(4×1) works and gives a 3×1 column, one total per shop. Option B fails the inner check: (3×4)·(1×3) puts 4 against 1. Options C and D contain no prices at all, so they cannot produce revenue.
Quiz
The boss asks: "how much MORE did each counter earn in February than January?" Which operation chain answers it?
- Multiply each month by the price column, then subtract
- Add the two sales matrices
- Multiply the two sales matrices together
- Take the inverse of the February matrix
Show the answer
Multiply each month by the price column, then subtract
"Earn" means revenue, so price each month first (S·P twice), then subtract the revenue columns: (171-132, 188-156) = (39, 32). Adding gives totals not growth, and multiplying two sales grids has no business meaning here: their columns are items, not prices.
Watch out
Where business matrices go wrong
Order of multiplication: S·P works, but P·S is (2×1)·(2×2), inner numbers 1 ≠ 2, not defined. Sales grid first, price column second.
Shape drift: if February adds a new item, a 2×2 cannot be added to a 2×3. Rebuild both months to the same shape (put 0s for the missing item) before adding.
Theory
This is a real industry pattern
Excel's SUMPRODUCT is one row-into-column. Inventory systems, GST billing software and banking ledgers all store quantity and rate tables and multiply them exactly like Rupa. Economists model entire countries with input-output matrices. In Semester 3 you will store graphs as adjacency matrices: same grid, new costume.
Summary
Key takeaways
- Model business data with a fixed convention: rows are things, columns are facts.
- Addition combines periods; subtraction shows growth.
- Sales grid × price column = revenue per row (shop, counter, branch).
- Scalar multiplication applies one uniform change to every cell.
- Check orders before every move: same order to add, inner match to multiply.
- Memory hook: rows are things, columns are facts.