Math Problem Statement
Solve the following system of linear equations using Matrices as well as Cramer’s rule.
2x – y + z = 5 4x +2y + 3z = 8 3x – 4y – z = 3
Solution
System of Linear Equations:
We are given the system:
2x - y + z &= 5 \\ 4x + 2y + 3z &= 8 \\ 3x - 4y - z &= 3 \end{align*}$$ #### Step 1: Write the system in matrix form. This system can be written as: $$A \cdot X = B$$ Where: $$A = \begin{pmatrix} 2 & -1 & 1 \\ 4 & 2 & 3 \\ 3 & -4 & -1 \end{pmatrix}, \quad X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad B = \begin{pmatrix} 5 \\ 8 \\ 3 \end{pmatrix}$$ ### Solution using Matrices (Inverse Method): To solve using matrices, we need to find $$X = A^{-1} \cdot B$$, where $$A^{-1}$$ is the inverse of matrix $$A$$. 1. **Find the determinant of $$A$$**. 2. **Find the inverse of matrix $$A$$**. 3. **Multiply the inverse matrix $$A^{-1}$$ by matrix $$B$$**. Let's proceed to calculate these steps. #### Step 2: Calculate the determinant of matrix $$A$$. The determinant of a 3x3 matrix $$A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}$$ is: $$\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)$$ For our matrix $$A$$: $$A = \begin{pmatrix} 2 & -1 & 1 \\ 4 & 2 & 3 \\ 3 & -4 & -1 \end{pmatrix}$$ Let's compute this determinant. $$\text{det}(A) = 2 \left( (2 \cdot -1) - (3 \cdot -4) \right) - (-1) \left( (4 \cdot -1) - (3 \cdot 3) \right) + 1 \left( (4 \cdot -4) - (2 \cdot 3) \right)$$ Now, solving for the determinant. #### Step 3: Solve using Cramer's Rule. Cramer's Rule states that the solution to a system of linear equations can be found using determinants. The solution for $$x, y, z$$ is: $$x = \frac{\text{det}(A_x)}{\text{det}(A)}, \quad y = \frac{\text{det}(A_y)}{\text{det}(A)}, \quad z = \frac{\text{det}(A_z)}{\text{det}(A)}$$ Where $$A_x, A_y, A_z$$ are matrices formed by replacing the columns of $$A$$ with matrix $$B$$. I'll calculate these steps. Let me proceed to compute the necessary determinants.### Solution Using Matrix Inverse: The determinant of matrix $$A$$ is: $$\text{det}(A) = -15$$ Since the determinant is non-zero, we can compute the inverse of $$A$$. The solution for the system using the inverse matrix is: $$\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 0.333 \\ -1.267 \\ 3.067 \end{pmatrix}$$ Thus, the solution is: $$x = 0.333, \quad y = -1.267, \quad z = 3.067$$ ### Solution Using Cramer's Rule: The determinants for Cramer's rule are: $$\text{det}(A_x) = 5, \quad \text{det}(A_y) = 19, \quad \text{det}(A_z) = -46$$ Using Cramer's rule: $$x = \frac{\text{det}(A_x)}{\text{det}(A)} = \frac{5}{-15} = 0.333$$ $$y = \frac{\text{det}(A_y)}{\text{det}(A)} = \frac{19}{-15} = -1.267$$ $$z = \frac{\text{det}(A_z)}{\text{det}(A)} = \frac{-46}{-15} = 3.067$$ The solutions for both methods match: $$x = 0.333, \quad y = -1.267, \quad z = 3.067$$ Would you like further details or have any questions? ### Related Questions: 1. How do you calculate the inverse of a 3x3 matrix? 2. What is the significance of a non-zero determinant in a matrix? 3. Can you explain how to derive Cramer's rule for 3 variables? 4. What would happen if the determinant of $$A$$ was zero? 5. How does Cramer's rule apply to systems with more than three variables? ### Tip: When the determinant of a matrix is zero, the system of equations either has no solution or infinitely many solutions, making the system singular.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Matrix Inversion
Cramer's Rule
Formulas
Matrix form: A ⋅ X = B
Determinant of a 3x3 matrix: det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
Cramer's Rule: x = det(A_x)/det(A), y = det(A_y)/det(A), z = det(A_z)/det(A)
Theorems
Cramer's Rule
Properties of Inverse Matrices
Suitable Grade Level
Grade 10-12
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