A finance company has offices located in ewery division,
every didtrict 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 superntendent, two clerks,
one typist and one poen. A district office, has in
addition, one clerk and one peon. The basic monthly
salaries are as follows :
Office superintendernt Rs 500, Head clerk Rs 200, cashier Rs 175, clerks and typist
Rs 150 and peon Rs 100. Using matrix motation find
the total basic monthly salary bill of all the offices taken
together.

Explanation

To solve the problem of calculating the total basic monthly salary bill of all the offices using matrix notation, we will follow these steps:

### Step 1: Define the Salaries
We start by defining the salaries of each position:
- Office Superintendent (OS): Rs 500
- Head Clerk (HC): Rs 200
- Cashier (C): Rs 175
- Clerk (Cl): Rs 150
- Typist (T): Rs 150
- Peon (P): Rs 100

### Step 2: Define the Structure of Each Office
Next, we define the number of each position in the different types of offices:
1. **Divisional Office**:
- OS: 1
- HC: 1
- C: 1
- Cl: 3 (1 + 2 additional)
- T: 1
- P: 1

2. **District Office**:
- OS: 0
- HC: 1
- C: 1
- Cl: 2 (1 + 1 additional)
- T: 0
- P: 2 (1 + 1 additional)

3. **Taluka Office**:
- OS: 0
- HC: 1
- C: 1
- Cl: 1
- T: 0
- P: 1

### Step 3: Create the Matrix for Each Office Type
We can represent the number of each position in a matrix form:

\[ \text{Office Matrix} = \begin{bmatrix} 1 & 1 & 1 & 3 & 1 & 1 \\ % Divisional Office 0 & 1 & 1 & 2 & 0 & 2 \\ % District Office 0 & 1 & 1 & 1 & 0 & 1 % Taluka Office \end{bmatrix} \]

### Step 4: Create the Salary Vector
Next, we create a salary vector:

\[ \text{Salary Vector} = \begin{bmatrix} 500 \\ % OS 200 \\ % HC 175 \\ % C 150 \\ % Cl 150 \\ % T 100 % P \end{bmatrix} \]

### Step 5: Perform Matrix Multiplication
Now, we multiply the office matrix by the salary vector to find the total salary for each type of office:

\[ \text{Total Salary} = \text{Office Matrix} \times \text{Salary Vector} \]

Calculating each row:
1. **Divisional Office**:
\[ 1 \times 500 + 1 \times 200 + 1 \times 175 + 3 \times 150 + 1 \times 150 + 1 \times 100 = 500 + 200 + 175 + 450 + 150 + 100 = 1575 \]

2. **District Office**:
\[ 0 \times 500 + 1 \times 200 + 1 \times 175 + 2 \times 150 + 0 \times 150 + 2 \times 100 = 0 + 200 + 175 + 300 + 0 + 200 = 875 \]

3. **Taluka Office**:
\[ 0 \times 500 + 1 \times 200 + 1 \times 175 + 1 \times 150 + 0 \times 150 + 1 \times 100 = 0 + 200 + 175 + 150 + 0 + 100 = 625 \]

### Step 6: Total Salary Calculation
Now we have the total salaries for each office type:
- Divisional Office: Rs 1575
- District Office: Rs 875
- Taluka Office: Rs 625

Next, we calculate the total salary bill for all offices:
- Number of Divisions: 5
- Number of Districts: 30
- Number of Talukas: 200

Total Salary Bill:
\[ \text{Total Salary Bill} = (1575 \times 5) + (875 \times 30) + (625 \times 200) \]

Calculating each term:
1. Divisional Offices: \(1575 \times 5 = 7875\)
2. District Offices: \(875 \times 30 = 26250\)
3. Taluka Offices: \(625 \times 200 = 125000\)

Adding them together:
\[ \text{Total Salary Bill} = 7875 + 26250 + 125000 = 153125 \]

### Final Answer
The total basic monthly salary bill of all the offices taken together is **Rs 153125**.

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