Q.The bookshop of a particular school has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economics books. Their selling prices are ₹80, ₹60 and ₹40 each respectively. Find the total amount the bookshop will receive from selling all the books using matrix algebra.

The key idea is Matrix Multiplication Compatibility: if we arrange the quantities and prices as matrices, the total revenue is the product of a row matrix (quantities) and a column matrix (prices).

Step 1: Convert dozens to individual units.

Chemistry: $10 \times 12 = 120$ books

Physics: $8 \times 12 = 96$ books

Economics: $10 \times 12 = 120$ books

Step 2: Write the quantity row matrix $Q$ and price column matrix $P$.

$Q = \begin{bmatrix} 120 & 96 & 120 \end{bmatrix}$, …

The problem gives us two sets of numbers: the quantities of books (in dozens) and the prices per book. We want the total money received. Matrix algebra lets us do this in one clean multiplication — but only if the matrices are compatible.

Why matrix multiplication works here:

If we arrange the quantities as a row matrix and the prices as a column matrix, multiplying them gives a single number (a $1 \times 1$ matrix). Each entry in the row multiplies the corresponding entry in the column, and we add the products. That’s exactly what “total amount” means:

$(\text{quantity of chemistry}) \times (\text{price of chemistry}) + (\text{quantity of physics}) \times (\text{price of physics}) + (\text{quantity of economics}) \times (\text{price of economics})$.

Let’s work through it step by step.

1. Convert dozens to individual books.

   Chemistry: $10$ dozen $= 10 \times 12 = 120$ books.

   Physics: $8$ dozen $= 8 \times 12 = 96$ books.

   Economics: $10$ dozen $= 10 \times 12 = 120$ books.

2. Write the quantity matrix as a row vector.

   We put the numbers in the same order as the prices will appear: chemistry, physics, economics.

This is a $1 \times 3$ matrix.

3. Write the price matrix as a column vector. Prices: chemistry ₹80, physics ₹60, economics ₹40.

This is a $3 \times 1$ matrix.

4. Check compatibility for multiplication. …

Method: Using a row-by-column product to total a priced inventory

When you need the total value of several items with known quantities and prices, arrange the quantities as a row matrix and the prices as a column matrix; their product is the single total.

Steps

Step 1: Put all quantities in consistent units

Convert any "dozens" to actual counts by multiplying by 12 before anything else.

Step 2: Build the row and column matrices …

Common Mistakes

Mistake 1: Forgetting to convert dozens to individual books

Why it's wrong: "10 dozen" is $10\times12=120$ books; multiplying the price by 10 instead of 120 undercounts the revenue massively. Correct approach: convert every dozen figure to a unit count first.

Mistake 2: Ordering the matrices so the product is the wrong shape …