Question

In a certain city there are 30 colleges, has 15 peons, 6 clerks, 1 typist, and 1 section officer. Express the given information as a column matrix. Using scalar multiplication, find the total number of posts of each kind in all the colleges.

Answer 1

Hint: We write the number of posts of each kind in one college. Form a matrix containing the elements as the number of posts in 1 college and another matrix with an element as the number of colleges in the city. After that use matrix multiplication to find the number of posts of each kind.

Complete step by step answer:
A matrix is an array of the given data arranged in the form of rows and columns. A matrix has an order $m \times n$ where m is the number of rows and n is the number of columns.
The total number of colleges in the city is 30.
We write the number of colleges in the city as a matrix with one element.
Let $A = \left[ {30} \right]$
The number of posts for peons in a college is 15.
The number of posts for clerks in a college is 6.
The number of posts for typists in a college is 1.
The number of posts for section officers in a college is 1.
We form a column matrix in which the elements are the number of posts of each kind.
Let the column matrix be B.
Then we can represent matrix B as:
Since we know the matrix B represents the number of posts of each kind in one college and matrix A represents the number of colleges in the city, we can find the total number of posts in 30 colleges by matrix multiplication of A and B.
We know in matrix multiplication the product matrix exists if and only if the number of columns in the first matrix is equal to the number of rows of the second matrix.
The general order of the matrix is written as $m \times n$ where m is the number of rows and n is the number of columns.
Here B is a $4 \times 1$ matrix and A is $1 \times 1$.
Since the number of rows of B is equal to the number of columns in A i.e.1 then the product matrix exists.
$ \Rightarrow B \times A = \left[ {\begin{array}{*{20}{c}}
{15}\\
6\\
1\\
1
\end{array}} \right] \times \left[ {30} \right]$
Multiply each element of B with a single element of A
$ \Rightarrow B \times A = \left[ {\begin{array}{*{20}{c}}
{15 \times 30}\\
{6 \times 30}\\
{1 \times 30}\\
{1 \times 30}
\end{array}} \right]$
Calculate the multiplication of elements inside the matrix.
$\therefore B \times A = \left[ {\begin{array}{*{20}{c}}
{4550}\\
{180}\\
{30}\\
{30}
\end{array}} \right]$

Hence, the total number of posts of peon in 30 colleges is 450, of the clerk is 180, of the typist is 30 and of section officer is 30.

Note: Students might make the mistake of representing the matrix multiplication as $A \times B$ which is incorrect because the number of columns of A is not equal to the number of rows of B. Always check if the product matrix exists or not and then perform the matrix multiplication.