Question

To promote the making of toilets for women, an organization tried to generate awareness through: 1) phone calls, 2) letters and 3) announcements. The cost for each mode per attempt is given by 1) Rs 50, 2) Rs 20, and 3) Rs 40. The number of attempts made in 3 villages X, Y, Z are given as
\[\begin{matrix}
   {} & (1) & (2) & (3) \\
   X & 400 & 300 & 100 \\
   Y & 300 & 250 & 75 \\
   Z & 500 & 400 & 150 \\
\end{matrix}\]
Now find the total cost incurred by the organization for the three villages separately using matrices.
Write one value generated by the organization in the society.

Answer 1

Hint: Here we can create two matrices, one which is a column which determines cost of each method of awareness. Second matrix will be a 3 × 3 matrix which is the given data where Row 1, Row 2 and Row 3 represents village X, village Y and village Z respectively and Column 1, Column 2, Column 3, represents number of attempts made by 1) phone calls, 2) letters and 3) announcements. Hence A × B will give us a row matrix in which each row gives the cost of each village X, Y and Z respectively.

Complete step-by-step answer:
Now let A be a matrix such that $A=[\begin{matrix}
   \text{cost of (1)} & \text{cost of (2)} & \text{cost of (3)} \\
\end{matrix}]$
Hence we get \[A=[\begin{matrix}
   50 & 20 & 40 \\
\end{matrix}]\]
Let matrix B be $B=\left[ \begin{matrix}
   400 & 300 & 100 \\
   300 & 250 & 75 \\
   500 & 400 & 150 \\
\end{matrix} \right]$
Now Row 1, Row 2 and Row 3 represents village X, village Y and village Z respectively and Column 1, Column 2, Column 3, represents number of attempts made by 1) phone calls, 2) letters and 3) announcements
Hence now A × B will be a 1 × 3 matrix in which each row represents total cost in each Village.
That is Column 1 will represent the total cost of village X.
That is Column 2 will represent the total cost of village Y.
That is Column 3 will represent the total cost of village Z.
Now let us find A × B
\[A\times B=[\begin{matrix}
   50 & 20 & 40 \\
\end{matrix}]\times \left[ \begin{matrix}
   400 & 300 & 100 \\
   300 & 250 & 75 \\
   500 & 400 & 150 \\
\end{matrix} \right]\]
\[\left[ \begin{matrix}
   20000+6000+4000 \\
   15000+5000+8000 \\
   25000+8000+6000 \\
\end{matrix} \right] \\
 =\left[ \begin{align}
  & 30000 \\
 & 23000 \\
 & 39000 \\
\end{align} \right] \\
\]
Hence Cost of X is 30000, cost of Y is 23000 and cost of Z is 39000.

Note: Now note that in matrix multiplication we multiply a $m\times n$ matrix to $n\times p$ matrix hence the number of rows of the second matrix must match the number of columns of the first matrix. Hence we should take the first matrix accordingly. In our case we took a 1 × 3 matrix.