A company has two production departments, P_{1} and P_{z} and three service departments S_{1} S_{2} and The direct cost allocated to each of the departments and the percentage of total cost of each service department apportioned to various departments are given below: S_{3}
Department
Direct Cost (2)
60,000
74,000
6000
8000
68,500
Percentage Allocation of Total Cost
40
35
25
20
40
20
0
20
35
15
0
20
25
5
0
Determine the total cost (allocated and apportioned) for each production department by using matrix algebra

To solve the problem using matrix algebra, let's break down the data and approach step-by-step.

### Step 1: Define the Matrices

1. **Direct Costs Vector**: This vector contains the direct costs for each department.
   \[
   \text{Direct Cost} = \begin{bmatrix}
   60,000 \\
   74,000 \\
   6,000 \\
   8,000 \\
   68,500
   \end{bmatrix}
   \]

2. **Percentage Allocation Matrix**: This matrix contains the percentage allocation of each service department's cost to the other departments. The departments are \(P_1\), \(P_2\), \(S_1\), \(S_2\), and \(S_3\) respectively.

   \[
   A = \begin{bmatrix}
   1 & 0 & -0.4 & -0.2 & -0.35 \\
   0 & 1 & -0.35 & -0.4 & -0.25 \\
   0 & 0 & 0.6 & -0.2 & -0.2 \\
   0 & 0 & -0.15 & 0.4 & -0.25 \\
   0 & 0 & -0.25 & -0.2 & 0.75
   \end{bmatrix}
   \]

### Step 2: Formulate the Matrix Equation

The total costs for each department, including both direct costs and costs apportioned from service departments, can be represented by the vector \( \textbf{x} \), which is the solution to the following matrix equation:

\[
A \times \textbf{x} = \textbf{b}
\]

Where \( \textbf{b} \) is the vector of direct costs.

### Step 3: Solve for \( \textbf{x} \)

To find \( \textbf{x} \), which represents the total costs for each department, we solve:

\[
\textbf{x} = A^{-1} \times \textbf{b}
\]

Where \( A^{-1} \) is the inverse of the matrix \( A \).

Let's calculate this step-by-step.

### Step 4: Calculate the Inverse of Matrix \( A \)

Using matrix algebra, find the inverse \( A^{-1} \).

### Step 5: Multiply the Inverse by the Direct Costs Vector \( \textbf{b} \)

This multiplication will yield the total costs for each department.

I will now calculate these values.The total costs (allocated and apportioned) for each department are as follows:

- \( P_1 \):  \( 215,839.89 \)
- \( P_2 \):  \( 241,831.46 \)
- \( S_1 \):  \( 133,988.76 \)
- \( S_2 \):  \( 186,292.13 \)
- \( S_3 \):  \( 185,674.16 \)

These values represent the total costs that each department incurs, including both direct costs and costs apportioned from the service departments.

Would you like more details on any step or explanation on how this calculation works?

Here are 5 questions that could help you expand on this topic:

1. How is the inverse of a matrix calculated, and what is its significance in solving linear equations?
2. What are some alternative methods to allocate service department costs to production departments?
3. How would changes in the allocation percentages affect the total cost for each department?
4. Can this matrix algebra approach be applied to a larger number of departments, and if so, how does the complexity scale?
5. What are some common pitfalls when using matrix algebra for cost allocation?

**Tip:** Understanding how matrix inversion works is crucial when solving systems of linear equations in various applications, including economics, engineering, and physics.

Learn how matrix algebra is applied to allocate costs between production and service departments. Solve a specific problem involving direct costs and percentage allocation, using matrix multiplication and inversion. Detailed step-by-step solution provided.

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