Q.A manufacturer produces three products $x, y, z$ which he sells in two markets. Annual sales are indicated below: Market I — $x$: 10,000, $y$: 2,000, $z$: 18,000 Market II — $x$: 6,000, $y$: 20,000, $z$: 8,000

Concept: Matrix Multiplication Compatibility – the number of columns in the first matrix must equal the number of rows in the second.

Step 1: Write the sales matrix $S$ (markets × products) and the price vector $P$ (products × 1).

Step 2: Revenue in each market = $S \cdot P$.

Step 3: Gross profit = Revenue – Cost. Cost vector $C = \begin{bmatrix}2.00 \\ 1.00 \\ 0.50\end{bmatrix}$. Total cost in each market = $S \cdot C$. …

Multiplying a row of quantities by a column of prices sums each product's quantity times its price, so stacking both markets into one matrix gives all the totals in a single product.

1. Sales matrix (rows = Market I, Market II; columns = $x,y,z$):

1. Total revenue in each market. Price column $P = \begin{bmatrix} 2.50 \\ 1.50 \\ 1.00 \end{bmatrix}$.

   $R = SP = \begin{bmatrix} 10000(2.50)+2000(1.50)+18000(1.00) \\ 6000(2.50)+20000(1.50)+8000(1.00) \end{bmatrix} = \begin{bmatrix} 25000+3000+18000 \\ 15000+30000+8000 \end{bmatrix} = \begin{bmatrix} 46000 \\ 53000 \end{bmatrix}$

   So revenue is $\text{₹}\,46{,}000$ in Market I and $\text{₹}\,53{,}000$ in Market II (total $\text{₹}\,99{,}000$).
2. Gross profit. Cost column $C = \begin{bmatrix} 2.00 \\ 1.00 \\ 0.50 \end{bmatrix}$ (50 paise $=\text{₹}\,0.50$). …

Method: Modelling revenue / cost / profit with a quantity–price matrix product

Use this for word problems where quantities and unit prices (or costs) must be combined "with the help of matrix algebra".

Steps

Step 1: Arrange quantities as a matrix and prices as a column.

Put the sales quantities in a matrix $S$ (one row per market/seller, one column per product) and the unit prices in a column vector $P$ (one entry per product). The inner dimensions must match so $SP$ is defined.

Step 2: Multiply $S \times P$ to get a total per row. …

Common Mistakes

Mistake 1: Multiplying in an order whose shapes don't line up.

Why it's wrong: to get a per-market total you need (markets $\times$ products)(products $\times$ 1); reversing the factors is undefined or gives the wrong meaning. Correct approach: check that the number of products (columns of $S$) equals the rows of the price column.

Mistake 2: Not converting paise to rupees consistently. …