Question

In a city there are two factories A and B. Each factory produces sports clothes for boys and girls. There are three types of clothes produced in both the factories, type I, II and III are 80, 70 and 65 in factory A and 85, 65 and 72 in factory B. For girls the number of units of I, II and III are 80, 75 and 90 in factory A and 50, 55, 80 in factory B. Express this information in terms of matrices and using matrix algebra answer the following questions.
1. Total units of type I produced for boys?
2. Total production of each type for boys?
3. Total production of each type for girls?

Answer 1

This problem deals with the basic concepts of matrices. Once you know how to arrange the given data in a matrix form then the rest of the job is very easy and it is basically the computation of the addition of the matrices arranged from the given data. We should be familiar with the basic arithmetic involved with the matrices.

Complete step-by-step solution:
Given that there are two factories named A and B and both the factories produce sport clothes for boys as well as girls.
So in each factory there are three types of sports clothes produced namely type I, type II, type III for both boys and girls.
Now representing the types of clothes for boys and girls in matrix form from each factory A and B.
Representing now the matrix of sports clothes produced in factory A, for both boys and girls, where there are 2 rows and 3 columns for which the 3 columns are for the three types of sports clothes produced whereas the 2 rows are for, one for boys and another for girls, as given below:
Factory A = \left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {80} \\\ {80} \end{array}}&{\begin{array}{*{20}{c}} {70} \\\ {75} \end{array}}&{\begin{array}{*{20}{c}} {65} \\\ {90} \end{array}} \end{array}} \right]
Representing now the matrix of sports clothes produced in factory B, for both boys and girls, where there are 2 rows and 3 columns for which the 3 columns are for the three types of sports clothes produced whereas the 2 rows are for, one for boys and another for girls, as given below:
Factory B = \left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {85} \\\ {50} \end{array}}&{\begin{array}{*{20}{c}} {65} \\\ {55} \end{array}}&{\begin{array}{*{20}{c}} {72} \\\ {80} \end{array}} \end{array}} \right]
Now finding the total no. of units of type I produced for boys, is given by:
We represented the first three columns to be type I, type II and type III respectively.
Hence the first column in both the factories A and B represents the type I production.
But here we have to find the type I production only for boys, we know that the first row in each matrix represents the production for boys of all types.
Hence the total no. of units of type I produced for boys is given by the sum of elements of first row and first column of both the matrices A and B, which is given by:
$\Rightarrow 80 + 85$
$\Rightarrow 165$
(2) Total production of each type for boys?
Here the first three columns of each matrix are type I, type II and type III production respectively.
The first row in each matrix represents the production for boys of all types.
Hence the production of each type for boys is the sum of the elements of the first row in matrix A plus the elements of the first row in matrix B, which is given by:
$\Rightarrow \left[ {80 + 70 + 65} \right] + \left[ {85 + 65 + 72} \right]$
$\Rightarrow \left[ {215} \right] + \left[ {222} \right]$
$\Rightarrow 437$
(3) Total production of each type for girls?
Here the first three columns of each matrix are type I, type II and type III production respectively.
The second row in each matrix represents the production for girls of all types.
Hence the production of each type for girls is the sum of the elements of the second row in matrix A plus the elements of the second row in matrix B, which is given by:
$\Rightarrow \left[ {80 + 75 + 90} \right] + \left[ {50 + 55 + 80} \right]$
$\Rightarrow \left[ {245} \right] + \left[ {185} \right]$
$\Rightarrow 430$

Total units of type I produced for boys is 165, Total production of each type for boys is 437 and the total production of each type for girls is 430.

Note: Here while solving or computing anything involved with matrices, here we should not get confused while computing the total no of units of production. Here when asked for a particular type of production then we go to the arranged column or row assigned to that particular production type and then start computing whatever is required.