1. **State the problem:** Solve the system of equations using the matrix inversion method: $$\begin{cases} x + 2y + 4z = 14 \\ 2x - y + 5z = 15 \\ -3x + 2y + 4z = 13 \end{cases}$$ 2. **Write the system in matrix form:** $$AX = B$$ where $$A = \begin{bmatrix} 1 & 2 & 4 \\ 2 & -1 & 5 \\ -3 & 2 & 4 \end{bmatrix}, \quad X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \quad B = \begin{bmatrix} 14 \\ 15 \\ 13 \end{bmatrix}$$ 3. **Find the inverse of matrix $A$, denoted $A^{-1}$:** Calculate the determinant $\det(A)$: $$\det(A) = 1((-1)(4) - 5(2)) - 2(2(4) - 5(-3)) + 4(2(2) - (-1)(-3))$$ $$= 1(-4 - 10) - 2(8 + 15) + 4(4 - 3) = 1(-14) - 2(23) + 4(1) = -14 - 46 + 4 = -56$$ Since $\det(A) \neq 0$, $A$ is invertible. 4. **Compute the adjugate matrix $\text{adj}(A)$ and then $A^{-1} = \frac{1}{\det(A)} \text{adj}(A)$:** Calculate cofactors and transpose to get adjugate: $$\text{adj}(A) = \begin{bmatrix} -14 & -46 & 4 \\ -23 & 16 & 6 \\ 15 & 2 & -5 \end{bmatrix}^T = \begin{bmatrix} -14 & -23 & 15 \\ -46 & 16 & 2 \\ 4 & 6 & -5 \end{bmatrix}$$ Therefore, $$A^{-1} = \frac{1}{-56} \begin{bmatrix} -14 & -23 & 15 \\ -46 & 16 & 2 \\ 4 & 6 & -5 \end{bmatrix} = \begin{bmatrix} \frac{1}{4} & \frac{23}{56} & -\frac{15}{56} \\ \frac{23}{28} & -\frac{2}{7} & -\frac{1}{28} \\ -\frac{1}{14} & -\frac{3}{28} & \frac{5}{56} \end{bmatrix}$$ 5. **Find $X$ by multiplying $A^{-1}$ and $B$:** $$X = A^{-1}B = \begin{bmatrix} \frac{1}{4} & \frac{23}{56} & -\frac{15}{56} \\ \frac{23}{28} & -\frac{2}{7} & -\frac{1}{28} \\ -\frac{1}{14} & -\frac{3}{28} & \frac{5}{56} \end{bmatrix} \begin{bmatrix} 14 \\ 15 \\ 13 \end{bmatrix}$$ Calculate each component: $$x = \frac{1}{4} \times 14 + \frac{23}{56} \times 15 - \frac{15}{56} \times 13 = 3.5 + 6.16 - 3.48 = 6.18$$ $$y = \frac{23}{28} \times 14 - \frac{2}{7} \times 15 - \frac{1}{28} \times 13 = 11.5 - 4.29 - 0.46 = 6.75$$ $$z = -\frac{1}{14} \times 14 - \frac{3}{28} \times 15 + \frac{5}{56} \times 13 = -1 - 1.61 + 1.16 = -1.45$$ 6. **Final solution:** $$x \approx 6.18, \quad y \approx 6.75, \quad z \approx -1.45$$ This means the values of $x$, $y$, and $z$ satisfy all three equations in the system.