Q.In a legislative assembly election, a political group hired a public relations firm to promote its candidate in three ways: telephone, house calls, and letters. The cost per contact (in paise) is given in matrix $A$ as $A = \begin{bmatrix} 40 \\ 100 \\ 50 \end{bmatrix}$ (Telephone, Housecall, Letter — Cost per contact). The number of contacts of each type made in two cities $X$ and $Y$ is given by $B = \begin{bmatrix} 1000 & 500 & 5000 \\ 3000 & 1000 & 10000 \end{bmatrix}$ (rows $X$, $Y$; columns Telephone, Housecall, Letter). Find the total amount spent by the group in the two cities $X$ and $Y$.

Concept: Matrix Multiplication Compatibility — the number of columns in the first matrix must equal the number of rows in the second. Here, $B$ is $2 \times 3$ and $A$ is $3 \times 1$, so $BA$ is defined and gives a $2 \times 1$ result (total cost per city).

Step 1: Write $B$ (contacts per city) and $A$ (cost per contact in paise).

$B = \begin{bmatrix} 1000 & 500 & 5000 \\ 3000 & 1000 & 10000 \end{bmatrix}$,

$A = \begin{bmatrix} 40 \\ 100 \\ 50 \end{bmatrix}$.

Step 2: Multiply $BA$:

For city $X$ (row 1): $(1000 \times 40) + (500 \times 100) + (5000 \times 50) = 40000 + 50000 + 250000 = 340000$ paise. …

The cost per contact (paise) is $A=\begin{bmatrix}40\\100\\50\end{bmatrix}$ and the contact counts are $B=\begin{bmatrix}1000&500&5000\\3000&1000&10000\end{bmatrix}$. The amount spent per city is $BA$:

$BA=\begin{bmatrix}1000(40)+500(100)+5000(50)\\[2pt]3000(40)+1000(100)+10000(50)\end{bmatrix}=\begin{bmatrix}40000+50000+250000\\120000+100000+500000\end{bmatrix}=\begin{bmatrix}340000\\720000\end{bmatrix}\text{ paise}.$ …

Method: Total-cost word problems via a counts $\times$ cost product

Use this when contact/quantity counts and per-unit costs are given as matrices and you need totals.

Steps

Step 1: Order the factors so the shapes align.

To get a per-city total you need (city $\times$ type)(type $\times$ 1), i.e. multiply the counts matrix $B$ by the cost column $A$, giving one total per city.

Step 2: Compute each row's dot product. …

Common Mistakes

Mistake 1: Multiplying in an incompatible order.

Why it's wrong: with $A$ a $3\times1$ column and $B$ a $2\times3$ counts matrix, only $BA$ ($2\times1$) is defined for a per-city total. Correct approach: check inner dimensions before multiplying.

Mistake 2: Forgetting the paise-to-rupee conversion.

Why it's wrong: costs are in paise, so the raw product is in paise; totals must be divided by $100$ for rupees. Correct approach: convert at the end, consistently. …