1. **State the problem:** Solve the system of linear equations using Cramer's rule: $$\begin{cases}-X_1 - 4X_2 + 2X_3 + X_4 = -32 \\ 2X_1 - X_2 + 7X_3 + 9X_4 = 14 \\ -X_1 + X_2 + 3X_3 + X_4 = 11 \\ X_1 - 2X_2 + X_3 - 4X_4 = -4 \end{cases}$$ 2. **Recall Cramer's rule:** For a system $AX = B$, where $A$ is the coefficient matrix, $X$ the variable vector, and $B$ the constants vector, the solution for each variable $X_i$ is given by: $$X_i = \frac{\det(A_i)}{\det(A)}$$ where $A_i$ is the matrix formed by replacing the $i$-th column of $A$ with $B$. 3. **Form the coefficient matrix $A$ and constant vector $B$: ** $$A = \begin{bmatrix}-1 & -4 & 2 & 1 \\ 2 & -1 & 7 & 9 \\ -1 & 1 & 3 & 1 \\ 1 & -2 & 1 & -4 \end{bmatrix}, \quad B = \begin{bmatrix}-32 \\ 14 \\ 11 \\ -4 \end{bmatrix}$$ 4. **Calculate $\det(A)$:** Calculate determinant of $A$ (using cofactor expansion or a calculator): $$\det(A) = -1 \times \det\begin{bmatrix}-1 & 7 & 9 \\ 1 & 3 & 1 \\ -2 & 1 & -4 \end{bmatrix} - (-4) \times \det\begin{bmatrix}2 & 7 & 9 \\ -1 & 3 & 1 \\ 1 & 1 & -4 \end{bmatrix} + 2 \times \det\begin{bmatrix}2 & -1 & 9 \\ -1 & 1 & 1 \\ 1 & -2 & -4 \end{bmatrix} + 1 \times \det\begin{bmatrix}2 & -1 & 7 \\ -1 & 1 & 3 \\ 1 & -2 & 1 \end{bmatrix}$$ Calculating each minor determinant: - $M_1 = \det\begin{bmatrix}-1 & 7 & 9 \\ 1 & 3 & 1 \\ -2 & 1 & -4 \end{bmatrix} = -1(3 \times -4 - 1 \times 1) - 7(1 \times -4 - 1 \times -2) + 9(1 \times 1 - 3 \times -2) = -1(-12 -1) -7(-4 + 2) + 9(1 + 6) = 13 + 14 + 63 = 90$ - $M_2 = \det\begin{bmatrix}2 & 7 & 9 \\ -1 & 3 & 1 \\ 1 & 1 & -4 \end{bmatrix} = 2(3 \times -4 - 1 \times 1) - 7(-1 \times -4 - 1 \times 1) + 9(-1 \times 1 - 3 \times 1) = 2(-12 -1) -7(4 -1) + 9(-1 -3) = -26 -21 -36 = -83$ - $M_3 = \det\begin{bmatrix}2 & -1 & 9 \\ -1 & 1 & 1 \\ 1 & -2 & -4 \end{bmatrix} = 2(1 \times -4 - 1 \times -2) - (-1)(-1 \times -4 - 1 \times 1) + 9(-1 \times -2 - 1 \times 1) = 2(-4 + 2) - (-1)(4 -1) + 9(2 -1) = 2(-2) - (-1)(3) + 9(1) = -4 + 3 + 9 = 8$ - $M_4 = \det\begin{bmatrix}2 & -1 & 7 \\ -1 & 1 & 3 \\ 1 & -2 & 1 \end{bmatrix} = 2(1 \times 1 - 3 \times -2) - (-1)(-1 \times 1 - 3 \times 1) + 7(-1 \times -2 - 1 \times 1) = 2(1 + 6) - (-1)(-1 -3) + 7(2 -1) = 2(7) - (-1)(-4) + 7(1) = 14 - 4 + 7 = 17$ Now substitute back: $$\det(A) = -1 \times 90 - (-4) \times (-83) + 2 \times 8 + 1 \times 17 = -90 - 332 + 16 + 17 = -389$$ 5. **Calculate determinants $\det(A_i)$ for each variable:** - Replace the $i$-th column of $A$ with $B$ and calculate determinant. **For $X_1$:** Replace first column: $$A_1 = \begin{bmatrix}-32 & -4 & 2 & 1 \\ 14 & -1 & 7 & 9 \\ 11 & 1 & 3 & 1 \\ -4 & -2 & 1 & -4 \end{bmatrix}$$ Calculate $\det(A_1)$ similarly: $$\det(A_1) = -32 \times \det\begin{bmatrix}-1 & 7 & 9 \\ 1 & 3 & 1 \\ -2 & 1 & -4 \end{bmatrix} - (-4) \times \det\begin{bmatrix}14 & 7 & 9 \\ 11 & 3 & 1 \\ -4 & 1 & -4 \end{bmatrix} + 2 \times \det\begin{bmatrix}14 & -1 & 9 \\ 11 & 1 & 1 \\ -4 & -2 & -4 \end{bmatrix} + 1 \times \det\begin{bmatrix}14 & -1 & 7 \\ 11 & 1 & 3 \\ -4 & -2 & 1 \end{bmatrix}$$ Calculate minors: - $M_1 = \det\begin{bmatrix}-1 & 7 & 9 \\ 1 & 3 & 1 \\ -2 & 1 & -4 \end{bmatrix} = 90$ (from before) - $M_2 = \det\begin{bmatrix}14 & 7 & 9 \\ 11 & 3 & 1 \\ -4 & 1 & -4 \end{bmatrix} = 14(3 \times -4 - 1 \times 1) - 7(11 \times -4 - 1 \times -4) + 9(11 \times 1 - 3 \times -4) = 14(-12 -1) -7(-44 +4) + 9(11 +12) = 14(-13) -7(-40) + 9(23) = -182 + 280 + 207 = 305$ - $M_3 = \det\begin{bmatrix}14 & -1 & 9 \\ 11 & 1 & 1 \\ -4 & -2 & -4 \end{bmatrix} = 14(1 \times -4 - 1 \times -2) - (-1)(11 \times -4 - 1 \times -4) + 9(11 \times -2 - 1 \times -4) = 14(-4 + 2) - (-1)(-44 + 4) + 9(-22 + 4) = 14(-2) - (-1)(-40) + 9(-18) = -28 - 40 - 162 = -230$ - $M_4 = \det\begin{bmatrix}14 & -1 & 7 \\ 11 & 1 & 3 \\ -4 & -2 & 1 \end{bmatrix} = 14(1 \times 1 - 3 \times -2) - (-1)(11 \times 1 - 3 \times -4) + 7(11 \times -2 - 1 \times -4) = 14(1 + 6) - (-1)(11 + 12) + 7(-22 + 4) = 14(7) - (-1)(23) + 7(-18) = 98 + 23 - 126 = -5$ Substitute: $$\det(A_1) = -32 \times 90 - (-4) \times 305 + 2 \times (-230) + 1 \times (-5) = -2880 - (-1220) - 460 - 5 = -2880 + 1220 - 460 - 5 = -2125$$ **For $X_2$:** Replace second column: $$A_2 = \begin{bmatrix}-1 & -32 & 2 & 1 \\ 2 & 14 & 7 & 9 \\ -1 & 11 & 3 & 1 \\ 1 & -4 & 1 & -4 \end{bmatrix}$$ Calculate $\det(A_2)$: $$\det(A_2) = -1 \times \det\begin{bmatrix}14 & 7 & 9 \\ 11 & 3 & 1 \\ -4 & 1 & -4 \end{bmatrix} - (-32) \times \det\begin{bmatrix}2 & 7 & 9 \\ -1 & 3 & 1 \\ 1 & 1 & -4 \end{bmatrix} + 2 \times \det\begin{bmatrix}2 & 14 & 9 \\ -1 & 11 & 1 \\ 1 & -4 & -4 \end{bmatrix} + 1 \times \det\begin{bmatrix}2 & 14 & 7 \\ -1 & 11 & 3 \\ 1 & -4 & 1 \end{bmatrix}$$ From previous calculations: - $\det\begin{bmatrix}14 & 7 & 9 \\ 11 & 3 & 1 \\ -4 & 1 & -4 \end{bmatrix} = 305$ - $\det\begin{bmatrix}2 & 7 & 9 \\ -1 & 3 & 1 \\ 1 & 1 & -4 \end{bmatrix} = -83$ Calculate new minors: - $M_3 = \det\begin{bmatrix}2 & 14 & 9 \\ -1 & 11 & 1 \\ 1 & -4 & -4 \end{bmatrix} = 2(11 \times -4 - 1 \times -4) - 14(-1 \times -4 - 1 \times 1) + 9(-1 \times -4 - 11 \times 1) = 2(-44 + 4) - 14(4 - 1) + 9(4 - 11) = 2(-40) - 14(3) + 9(-7) = -80 - 42 - 63 = -185$ - $M_4 = \det\begin{bmatrix}2 & 14 & 7 \\ -1 & 11 & 3 \\ 1 & -4 & 1 \end{bmatrix} = 2(11 \times 1 - 3 \times -4) - 14(-1 \times 1 - 3 \times 1) + 7(-1 \times -4 - 11 \times 1) = 2(11 + 12) - 14(-1 - 3) + 7(4 - 11) = 2(23) - 14(-4) + 7(-7) = 46 + 56 - 49 = 53$ Substitute: $$\det(A_2) = -1 \times 305 - (-32) \times (-83) + 2 \times (-185) + 1 \times 53 = -305 - 2656 - 370 + 53 = -3278$$ **For $X_3$:** Replace third column: $$A_3 = \begin{bmatrix}-1 & -4 & -32 & 1 \\ 2 & -1 & 14 & 9 \\ -1 & 1 & 11 & 1 \\ 1 & -2 & -4 & -4 \end{bmatrix}$$ Calculate $\det(A_3)$: $$\det(A_3) = -1 \times \det\begin{bmatrix}-1 & 14 & 9 \\ 1 & 11 & 1 \\ -2 & -4 & -4 \end{bmatrix} - (-4) \times \det\begin{bmatrix}2 & 14 & 9 \\ -1 & 11 & 1 \\ 1 & -4 & -4 \end{bmatrix} + (-32) \times \det\begin{bmatrix}2 & -1 & 9 \\ -1 & 1 & 1 \\ 1 & -2 & -4 \end{bmatrix} + 1 \times \det\begin{bmatrix}2 & -1 & 14 \\ -1 & 1 & 11 \\ 1 & -2 & -4 \end{bmatrix}$$ Calculate minors: - $M_1 = \det\begin{bmatrix}-1 & 14 & 9 \\ 1 & 11 & 1 \\ -2 & -4 & -4 \end{bmatrix} = -1(11 \times -4 - 1 \times -4) - 14(1 \times -4 - 1 \times -2) + 9(1 \times -4 - 11 \times -2) = -1(-44 + 4) - 14(-4 + 2) + 9(-4 + 22) = -1(-40) - 14(-2) + 9(18) = 40 + 28 + 162 = 230$ - $M_2 = \det\begin{bmatrix}2 & 14 & 9 \\ -1 & 11 & 1 \\ 1 & -4 & -4 \end{bmatrix} = -185$ (from above) - $M_3 = \det\begin{bmatrix}2 & -1 & 9 \\ -1 & 1 & 1 \\ 1 & -2 & -4 \end{bmatrix} = 8$ (from above) - $M_4 = \det\begin{bmatrix}2 & -1 & 14 \\ -1 & 1 & 11 \\ 1 & -2 & -4 \end{bmatrix} = 2(1 \times -4 - 11 \times -2) - (-1)(-1 \times -4 - 11 \times 1) + 14(-1 \times -2 - 1 \times 1) = 2(-4 + 22) - (-1)(4 - 11) + 14(2 - 1) = 2(18) - (-1)(-7) + 14(1) = 36 - 7 + 14 = 43$ Substitute: $$\det(A_3) = -1 \times 230 - (-4) \times (-185) + (-32) \times 8 + 1 \times 43 = -230 - 740 - 256 + 43 = -1183$$ **For $X_4$:** Replace fourth column: $$A_4 = \begin{bmatrix}-1 & -4 & 2 & -32 \\ 2 & -1 & 7 & 14 \\ -1 & 1 & 3 & 11 \\ 1 & -2 & 1 & -4 \end{bmatrix}$$ Calculate $\det(A_4)$: $$\det(A_4) = -1 \times \det\begin{bmatrix}-1 & 7 & 14 \\ 1 & 3 & 11 \\ -2 & 1 & -4 \end{bmatrix} - (-4) \times \det\begin{bmatrix}2 & 7 & 14 \\ -1 & 3 & 11 \\ 1 & 1 & -4 \end{bmatrix} + 2 \times \det\begin{bmatrix}2 & -1 & 14 \\ -1 & 1 & 11 \\ 1 & -2 & -4 \end{bmatrix} + (-32) \times \det\begin{bmatrix}2 & -1 & 7 \\ -1 & 1 & 3 \\ 1 & -2 & 1 \end{bmatrix}$$ Calculate minors: - $M_1 = \det\begin{bmatrix}-1 & 7 & 14 \\ 1 & 3 & 11 \\ -2 & 1 & -4 \end{bmatrix} = -1(3 \times -4 - 11 \times 1) - 7(1 \times -4 - 11 \times -2) + 14(1 \times 1 - 3 \times -2) = -1(-12 - 11) - 7(-4 + 22) + 14(1 + 6) = 23 - 126 + 98 = -5$ - $M_2 = \det\begin{bmatrix}2 & 7 & 14 \\ -1 & 3 & 11 \\ 1 & 1 & -4 \end{bmatrix} = 2(3 \times -4 - 11 \times 1) - 7(-1 \times -4 - 11 \times 1) + 14(-1 \times 1 - 3 \times 1) = 2(-12 - 11) - 7(4 - 11) + 14(-1 - 3) = 2(-23) - 7(-7) + 14(-4) = -46 + 49 - 56 = -53$ - $M_3 = \det\begin{bmatrix}2 & -1 & 14 \\ -1 & 1 & 11 \\ 1 & -2 & -4 \end{bmatrix} = 43$ (from above) - $M_4 = \det\begin{bmatrix}2 & -1 & 7 \\ -1 & 1 & 3 \\ 1 & -2 & 1 \end{bmatrix} = 17$ (from above) Substitute: $$\det(A_4) = -1 \times (-5) - (-4) \times (-53) + 2 \times 43 + (-32) \times 17 = 5 - 212 + 86 - 544 = -665$$ 6. **Calculate each variable:** $$X_1 = \frac{\det(A_1)}{\det(A)} = \frac{-2125}{-389} = \frac{2125}{389} \approx 5.46$$ $$X_2 = \frac{\det(A_2)}{\det(A)} = \frac{-3278}{-389} = \frac{3278}{389} \approx 8.42$$ $$X_3 = \frac{\det(A_3)}{\det(A)} = \frac{-1183}{-389} = \frac{1183}{389} \approx 3.04$$ $$X_4 = \frac{\det(A_4)}{\det(A)} = \frac{-665}{-389} = \frac{665}{389} \approx 1.71$$ **Final answer:** $$\boxed{X_1 = \frac{2125}{389}, \quad X_2 = \frac{3278}{389}, \quad X_3 = \frac{1183}{389}, \quad X_4 = \frac{665}{389}}$$