A finance company has offices located in every division,
every district and every taluka in a certain state in India.
Assume that there are five divisions, thirty districts and
200 talukas in the state. Each office has one head clerk,
one cashier, one clerk and one peon. A divisional office
has, in addition, one office superintendent, two clerks,
one typist and one peon. A district office, has in
addition, one clerk and one peon. The basic monthly
salaries are as follows :
Office superintendent Rs 500, Head clerk Rs 200, cashier Rs 175, clerks and typist
Rs 150 and peon Rs 100. Using matrix notation find
the total basic monthly salary bill of all the offices taken
together.

Answer

Total basic monthly salary bill of all the offices taken together
is Rs 159625.

Explanation

Lets us use the symbols Div, Dis, Tal for division, district,
taluka respectively and O, H, C, Cl , T and P for office
superintendent, Head clerk, Cashier, Clerk, Typist and Peon
respectively.
Then the number of offices can be arranged as elements of a
row matrix A and the composition of staffin various offices
can be areanged in a `3xx6` matrix B (say).
`{:(Div, Dis, Tal):}`
`therefore A = [[5,30,200]]`
and` B=[[1,1,1,2+1,1,1+1],[0,1,1,1+1,0,1+1],[0,1,1,1,0,1]]`
or `B= [[1,1,1,3,1,2],[0,1,1,2,0,2],[0,1,1,1,0,1]]`
The basic monthly salarise of various types of employees of
these offies correspond to the elements of the column matrix
C.
`thereforeC={:[O],[H],[C],[Cl],[T],[P]:}[[500],[200],[175],[150],[150],[100]]`
The total basic monthly salary bill of all the offices taken
together
`= A BC = A(BC)`
`= [ (5,30,200)]xx[[1675],[875],[625]]`
`[5xx1675+30xx865+200xx625]`
`= [159625]`
Hence, total basic monthly salary bill of all the offices
taken together is Rs 159625.