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 $\mathbf{A} \mathbf{I} = \mathbf{V}$: