(a) What is the matrix of funds collected by School B ?

OR

(b) What is the total amount of funds collected by Schools A and C ?

Q: Why does multiplying a 1×3 price matrix by a 3×1 quantity matrix give a single number?

By the matrix multiplication rule, a ($1 \times 3$) times a ($3 \times 1$) product has order $1 \times 1$ — a single entry equal to the sum of the paired products ($25\times30 + 50\times35 + 10\times50$). That single value is School B's total collection, Rs 3,000.

Matrices

Matrix Multiplication for Funds Collected

Official Answer

Each school's $\text{funds} = (\text{price of item} \times \text{quantity sold})$ , summed over fans, mats and plates.

Part (a) — Matrix of funds collected by School B

School B sold fans 30, mats 35, plates 50.

$\text{Funds}(B) = \begin{bmatrix}25 & 50 & 10\end{bmatrix} \times \begin{bmatrix}30 \\ 35 \\ 50\end{bmatrix}$
= $(25 \times 30) + (50 \times 35) + (10 \times 50)$
= $750 + 1750 + 500$
= $[3000]$

School B collected Rs 3,000 (a $1 \times 1$ matrix $[3000]$ ).

Part (b) — Total funds collected by Schools A and C

School A: $(25 \times 50) + (50 \times 60) + (10 \times 40) = 1250 + 3000 + 400$ = Rs 4,650.
School C: $(25 \times 35) + (50 \times 40) + (10 \times 25) = 875 + 2000 + 250$ = Rs 3,125.
Total $(A + C) = 4650 + 3125$ = Rs 7,775.

matrix multiplicationfunds collectedprice times quantityrow by columnSchool B Rs 3000Schools A and C Rs 7775total revenue matrix

Marking Scheme

1Part (a): 1 mark for setting up the product $\begin{bmatrix}25 & 50 & 10\end{bmatrix}\times\begin{bmatrix}30 \\ 35 \\ 50\end{bmatrix}$ ; 1 mark for the result Rs 3,000.
2Part (b): 1 mark for funds of A = Rs 4,650 and C = Rs 3,125; 1 mark for the total Rs 7,775.
3Accept the answer expressed as a $1 \times 1$ matrix or a plain rupee value.

Hint

$\text{Funds} = \text{price} \times \text{quantity}$ , summed over the three items. For (a) multiply the price row $\begin{bmatrix}25 & 50 & 10\end{bmatrix}$ by School B's quantity column $\begin{bmatrix}30 \\ 35 \\ 50\end{bmatrix}$ . For (b) do the same for School A and School C and add the two totals.

Quick Oral Answer

Multiplying the price row $\begin{bmatrix}25 & 50 & 10\end{bmatrix}$ by each school's quantity column gives the funds; School B raised Rs 3,000, and Schools A and C together raised $4650 + 3125$ = Rs 7,775.

Analysis & Explanation

This is the pay-off part of the case study, where price and sales matrices are combined by matrix multiplication.

Concept

Multiplying the $1 \times 3$ price row matrix by a $3 \times 1$ school-quantity column gives a $1 \times 1$ matrix whose single entry is that school's total collection — a direct application of the row-by-column rule of matrix multiplication.
For total funds of two schools, either add their individual products or multiply the price row by the sum of their two quantity columns.

Exam trap

Aligning the wrong price with the wrong item (order mismatch) is the top error; always pair 25↔fans, 50↔mats, 10↔plates.
In (b), students sometimes forget to add BOTH schools, reporting only one total.

Real-world

This is exactly how a retailer computes revenue per outlet from a shared price list and store-wise sales — a single matrix product replaces many manual multiplications.

Common Mistakes

1Pairing prices with the wrong items because the price and quantity orders were not aligned.
2In part (b), computing only one school's funds and forgetting to add the other.
3Arithmetic slips in the row-by-column sums (e.g. $50 \times 60 = 3000$ miscalculated).

Interesting Facts

The row-by-column multiplication rule used here is the same operation that powers everything from computer graphics transforms to Google's PageRank algorithm.

Total funds across all three schools (A 4650 + B 3000 + C 3125) come to Rs 10,775 — a neat way to check the individual computations.

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Frequently Asked Questions

Why does multiplying a 1×3 price matrix by a 3×1 quantity matrix give a single number?

By the matrix multiplication rule, a ( $1 \times 3$ ) times a ( $3 \times 1$ ) product has order $1 \times 1$ — a single entry equal to the sum of the paired products ( $25\times30 + 50\times35 + 10\times50$ ). That single value is School B's total collection, Rs 3,000.

How do I find the total funds of two schools together?

Compute each school's funds separately by the same price×quantity method and add them: School A = Rs 4,650 and School C = Rs 3,125, so together they collected Rs 7,775.