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: