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In this circuit, we are asked to write a matrix equation for the loop currents \( I_1, I_2, I_3, I_4 \) based on the given resistances and voltage sources. We will use Kirchhoff's Voltage Law (KVL) for each loop to set up the system of equations. This law states that the sum of the potential differences (voltage) in any closed loop must be zero.

### Loop Currents and Loops:

Let the loop currents \( I_1, I_2, I_3, I_4 \) correspond to the clockwise currents in the four loops, and we assume:

1. Loop 1 contains the current \( I_1 \),
2. Loop 2 contains \( I_2 \),
3. Loop 3 contains \( I_3 \),
4. Loop 4 contains \( I_4 \).

### Setting up the KVL Equations

We will sum the voltages in each loop, considering resistors shared between adjacent loops and the effect of voltage sources:

1. **For Loop 1**:
   \[
   20V - 6\Omega(I_1 - I_2) - 5\Omega I_1 = 0
   \]
   Simplified: 
   \[
   (6 + 5) I_1 - 6 I_2 = 20
   \]
   \[
   11 I_1 - 6 I_2 = 20
   \]

2. **For Loop 2**:
   \[
   34V - 6\Omega(I_2 - I_1) - 2\Omega(I_2 - I_3) = 0
   \]
   Simplified:
   \[
   -6 I_1 + (6 + 2) I_2 - 2 I_3 = 34
   \]
   \[
   -6 I_1 + 8 I_2 - 2 I_3 = 34
   \]

3. **For Loop 3**:
   \[
   19V - 2\Omega(I_3 - I_2) - 3\Omega(I_3 - I_4) = 0
   \]
   Simplified:
   \[
   -2 I_2 + (2 + 3) I_3 - 3 I_4 = 19
   \]
   \[
   -2 I_2 + 5 I_3 - 3 I_4 = 19
   \]

4. **For Loop 4**:
   \[
   23V - 3\Omega(I_4 - I_3) - 2\Omega I_4 = 0
   \]
   Simplified:
   \[
   -3 I_3 + (3 + 2) I_4 = 23
   \]
   \[
   -3 I_3 + 5 I_4 = 23
   \]

### Matrix Equation Form

We can now represent these equations in matrix form \( \mathbf{A} \mathbf{I} = \mathbf{V} \):

\[
\begin{pmatrix}
11 & -6  &  0  &  0  \\
-6 &  8  & -2  &  0  \\
0  & -2  &  5  & -3  \\
0  &  0  & -3  &  5
\end{pmatrix}
\begin{pmatrix}
I_1 \\
I_2 \\
I_3 \\
I_4
\end{pmatrix}
=
\begin{pmatrix}
20 \\
34 \\
19 \\
23
\end{pmatrix}
\]

### Final Matrix Form:

\[
\begin{pmatrix}
11 & -6  &  0  &  0  \\
-6 &  8  & -2  &  0  \\
0  & -2  &  5  & -3  \\
0  &  0  & -3  &  5
\end{pmatrix}
\begin{pmatrix}
I_1 \\
I_2 \\
I_3 \\
I_4
\end{pmatrix}
=
\begin{pmatrix}
20 \\
34 \\
19 \\
23
\end{pmatrix}
\]

This matrix equation can now be solved using methods such as Gaussian elimination or using matrix inversion to find the loop currents \( I_1, I_2, I_3, I_4 \).

---

Would you like details on solving this system or any other questions?

Here are some related questions for deeper understanding:

1. How can we apply Gaussian elimination to solve this matrix equation?
2. What is the physical significance of each term in the KVL equations?
3. How do we handle dependent loops with shared resistances?
4. What happens if we reverse the polarity of one of the voltage sources?
5. How can we confirm that our loop current directions are correct?

**Tip**: When solving electrical circuits using KVL, always ensure consistent direction for loop currents to avoid sign confusion.

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

Learn how to set up and solve a matrix equation to determine loop currents in a circuit using Kirchhoff's Voltage Law (KVL). Understand the principles of electrical circuit analysis and the application of linear algebra in determining currents in a closed loop. Suitable for undergraduate electrical engineering students.

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