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 number of posts in 1 college and another matrix with element as number of colleges in the city. Use the matrix multiplication to find the number of posts of each kind.
* 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 number of rows and n is number of columns.

Complete step-by-step answer:
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 \[A = \left[ {30} \right]\]
Number of posts for peons in a college is 15
Number of posts for clerks in a college is 6
Number of posts for typists in a college is 1
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 row of the second matrix.
General order of matrix is written as \[m \times n\]where m is number of rows and n is 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 number of columns in A i.e.1 then 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 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.
\[ \Rightarrow B \times A = \left[ {\begin{array}{*{20}{c}}
  {450} \\
  {180} \\
  {30} \\
  {30}
\end{array}} \right]\]
So, total number of posts of peon in 30 colleges is 450
Total number of posts of clerk in 30 colleges is 180
Total number of posts of typist in 30 colleges is 30
Total number of posts of section officer in 30 colleges 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.