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 number of posts of each kind in all the offices
taken together,

Explanation

To solve the problem of finding the total number of posts of each kind in all the offices of the finance company, we can use matrix notation. Let's break down the solution step by step.

### Step 1: Define the matrices

1. **Matrix A**: This matrix represents the number of offices in each category. We have:
- 5 Divisions
- 30 Districts
- 200 Talukas

Thus, we can represent this as a row matrix:
\[ A = \begin{bmatrix} 5 & 30 & 200 \end{bmatrix} \]

2. **Matrix B**: This matrix represents the number of each type of post in each office type. The posts are:
- Office Superintendent (O)
- Head Clerk (C)
- Cashier (Ca)
- Clerk (Cl)
- Typist (T)
- Peon (P)

The structure of matrix B is as follows:
- For Divisional Office: 1 O, 1 C, 1 Ca, 3 Cl, 1 T, 1 P
- For District Office: 0 O, 1 C, 1 Ca, 2 Cl, 0 T, 1 P
- For Taluka Office: 0 O, 0 C, 1 Ca, 0 Cl, 0 T, 1 P

Therefore, matrix B can be represented as:
\[ B = \begin{bmatrix} 1 & 1 & 1 & 3 & 1 & 1 \\ 0 & 1 & 1 & 2 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 & 1 \end{bmatrix} \]

### Step 2: Calculate the total number of posts

To find the total number of posts, we multiply matrix A by matrix B. The resulting matrix will give us the total number of each type of post across all offices.

\[ \text{Total Posts} = A \times B \]

### Step 3: Perform the matrix multiplication

Calculating the product \( A \times B \):

\[ \text{Total Posts} = \begin{bmatrix} 5 & 30 & 200 \end{bmatrix} \times \begin{bmatrix} 1 & 1 & 1 & 3 & 1 & 1 \\ 0 & 1 & 1 & 2 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 & 1 \end{bmatrix} \]

Calculating each element of the resulting matrix:

1. **For Office Superintendent (O)**:
\[ 5 \times 1 + 30 \times 0 + 200 \times 0 = 5 \]

2. **For Head Clerk (C)**:
\[ 5 \times 1 + 30 \times 1 + 200 \times 0 = 5 + 30 = 35 \]

3. **For Cashier (Ca)**:
\[ 5 \times 1 + 30 \times 1 + 200 \times 1 = 5 + 30 + 200 = 235 \]

4. **For Clerk (Cl)**:
\[ 5 \times 3 + 30 \times 2 + 200 \times 0 = 15 + 60 = 75 \]

5. **For Typist (T)**:
\[ 5 \times 1 + 30 \times 0 + 200 \times 0 = 5 \]

6. **For Peon (P)**:
\[ 5 \times 1 + 30 \times 1 + 200 \times 1 = 5 + 30 + 200 = 235 \]

Thus, the resulting matrix of total posts is:
\[ \text{Total Posts} = \begin{bmatrix} 5 & 35 & 235 & 75 & 5 & 235 \end{bmatrix} \]

### Final Result

The total number of posts in all the offices is:
- Office Superintendents: 5
- Head Clerks: 35
- Cashiers: 235
- Clerks: 75
- Typists: 5
- Peons: 235