1. **Problem Statement:** Solve the system of equations for the unknown currents $I_1$, $I_2$, and $I_3$ in the circuit from Exercise 5 using matrix methods. 2. **Set up the equations:** Using Kirchhoff's Voltage Law (KVL) and Ohm's Law, write the equations for the circuit loops. - Loop 1 (top-left loop): $$8 - 2I_1 - 2I_2 = 0$$ - Loop 2 (right loop): $$2I_2 - 4I_3 - 6 = 0$$ - Junction rule (current at node): $$I_1 + I_2 = I_3$$ 3. **Rewrite equations:** $$2I_1 + 2I_2 = 8$$ $$2I_2 - 4I_3 = 6$$ $$I_1 + I_2 - I_3 = 0$$ 4. **Express in matrix form:** $$\begin{bmatrix} 2 & 2 & 0 \\ 0 & 2 & -4 \\ 1 & 1 & -1 \end{bmatrix} \begin{bmatrix} I_1 \\ I_2 \\ I_3 \end{bmatrix} = \begin{bmatrix} 8 \\ 6 \\ 0 \end{bmatrix}$$ 5. **Solve the matrix equation:** Use Gaussian elimination or matrix inverse. - From equation 3: $$I_3 = I_1 + I_2$$ - Substitute into equation 2: $$2I_2 - 4(I_1 + I_2) = 6 \Rightarrow 2I_2 - 4I_1 - 4I_2 = 6 \Rightarrow -4I_1 - 2I_2 = 6$$ - Equation 1: $$2I_1 + 2I_2 = 8$$ 6. **Solve system:** From equation 1: $$I_1 + I_2 = 4$$ From modified equation 2: $$-4I_1 - 2I_2 = 6$$ Multiply equation 1 by 2: $$2I_1 + 2I_2 = 8$$ Add to equation 2: $$(-4I_1 - 2I_2) + (2I_1 + 2I_2) = 6 + 8 \Rightarrow -2I_1 = 14 \Rightarrow I_1 = -7$$ Substitute $I_1$ into $I_1 + I_2 = 4$: $$-7 + I_2 = 4 \Rightarrow I_2 = 11$$ Find $I_3$: $$I_3 = I_1 + I_2 = -7 + 11 = 4$$ 7. **Final answer:** $$I_1 = -7, \quad I_2 = 11, \quad I_3 = 4$$ Negative current $I_1$ indicates direction opposite to assumed.