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The circuit provided contains four loops with different resistors and voltage sources. To find the loop currents \( I_1, I_2, I_3, I_4 \) using Kirchhoff's Voltage Law (KVL), we need to apply the law to each loop and express the equations in matrix form.

Here is a step-by-step process for setting up the matrix equation:

### Step 1: Assign Loop Currents
Each loop has a current \( I_1, I_2, I_3, I_4 \), as indicated in the diagram.

### Step 2: Write KVL Equations
For each loop, sum the voltage drops across resistors and voltage sources, following Kirchhoff's Voltage Law (KVL). The voltage drops across each resistor are given by Ohm’s Law \( V = IR \), where \( R \) is the resistance, and \( I \) is the current.

1. **Loop 1 (Top Loop):**
   \[
   22 - 3I_1 - 2(I_1 - I_2) = 0
   \]
   Simplifying:
   \[
   22 = 5I_1 - 2I_2
   \]

2. **Loop 2:**
   \[
   6 - 4I_2 - 2(I_2 - I_1) - 3(I_2 - I_3) = 0
   \]
   Simplifying:
   \[
   6 = -2I_1 + 9I_2 - 3I_3
   \]

3. **Loop 3:**
   \[
   19 - 4I_3 - 3(I_3 - I_2) - 2(I_3 - I_4) = 0
   \]
   Simplifying:
   \[
   19 = -3I_2 + 9I_3 - 2I_4
   \]

4. **Loop 4 (Bottom Loop):**
   \[
   13 - 4I_4 - 2(I_4 - I_3) = 0
   \]
   Simplifying:
   \[
   13 = -2I_3 + 6I_4
   \]

### Step 3: Convert to Matrix Form
Now, arrange these equations in matrix form:

\[
\begin{pmatrix}
5 & -2 & 0 & 0 \\
-2 & 9 & -3 & 0 \\
0 & -3 & 9 & -2 \\
0 & 0 & -2 & 6
\end{pmatrix}
\begin{pmatrix}
I_1 \\
I_2 \\
I_3 \\
I_4
\end{pmatrix}
=
\begin{pmatrix}
22 \\
6 \\
19 \\
13
\end{pmatrix}
\]

This is the matrix equation that determines the loop currents.

### Next Steps:
- Solve this matrix equation using Gaussian elimination or matrix inversion to find the values of \( I_1, I_2, I_3, I_4 \).

Would you like to solve for the currents or need further clarifications?

### 5 Relative Questions:
1. How would you solve this matrix equation using Gaussian elimination?
2. What method would you use to solve this system in MATLAB or Python?
3. What happens to the loop currents if the resistances are doubled?
4. Can you apply a similar approach for a mesh analysis with more than four loops?
5. What physical interpretations can you draw from the values of the loop currents?

### Tip:
Always check for dependent loops or redundant loops in a circuit to simplify the system before solving.

Write a matrix equation that determines the loop currents for the given circuit with four loops, using Kirchhoff's Voltage Law (KVL).

This problem involves solving for loop currents in a four-loop circuit using Kirchhoff's Voltage Law (KVL). Learn how to apply KVL, Ohm’s Law, and matrix algebra to write and solve a system of linear equations for the currents in each loop. The circuit contains resistors and voltage sources, making it a practical example for electrical engineering students.

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