Math Problem Statement
Solve the equations by matrix inversion method: - x + y + z = 3; x + 2y + 3z = 4 ; x + 4y + 9z = 6
Solution
The given system of linear equations can be represented as:
1x + 1y + 1z &= 3 \\ 1x + 2y + 3z &= 4 \\ 1x + 4y + 9z &= 6 \end{aligned}$$ This system of equations can be written in matrix form as: $$A \cdot X = B$$ Where: $$A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 4 & 9 \end{bmatrix}, \quad X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \quad B = \begin{bmatrix} 3 \\ 4 \\ 6 \end{bmatrix}$$ ### Step 1: Find the inverse of matrix $$A$$ To solve for $$X$$, we need to calculate: $$X = A^{-1} \cdot B$$ We'll compute $$A^{-1}$$ (the inverse of matrix $$A$$) using standard methods like the adjugate formula or row reduction. ### Step 2: Compute determinant of $$A$$ The determinant of matrix $$A$$ is: $$\text{det}(A) = 1 \times (2 \times 9 - 4 \times 3) - 1 \times (1 \times 9 - 1 \times 3) + 1 \times (1 \times 4 - 1 \times 2)$$ $$\text{det}(A) = 1 \times (18 - 12) - 1 \times (9 - 3) + 1 \times (4 - 2) = 6 - 6 + 2 = 2$$ Since $$\text{det}(A) = 2$$, the matrix is invertible. ### Step 3: Find the cofactor matrix of $$A$$ Next, we find the cofactor matrix, which is the matrix of minors with signs: $$\text{Cofactor}(A) = \begin{bmatrix} 2 & -6 & 4 \\ -6 & 8 & -2 \\ 4 & -2 & 0 \end{bmatrix}$$ ### Step 4: Find the adjugate of $$A$$ The adjugate of $$A$$ is the transpose of the cofactor matrix: $$\text{Adj}(A) = \begin{bmatrix} 2 & -6 & 4 \\ -6 & 8 & -2 \\ 4 & -2 & 0 \end{bmatrix}^T = \begin{bmatrix} 2 & -6 & 4 \\ -6 & 8 & -2 \\ 4 & -2 & 0 \end{bmatrix}$$ ### Step 5: Compute the inverse of $$A$$ The inverse of $$A$$ is: $$A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{Adj}(A) = \frac{1}{2} \cdot \begin{bmatrix} 2 & -6 & 4 \\ -6 & 8 & -2 \\ 4 & -2 & 0 \end{bmatrix}$$ $$A^{-1} = \begin{bmatrix} 1 & -3 & 2 \\ -3 & 4 & -1 \\ 2 & -1 & 0 \end{bmatrix}$$ ### Step 6: Multiply $$A^{-1}$$ with $$B$$ Now, we multiply $$A^{-1}$$ with $$B$$ to find $$X$$: $$X = A^{-1} \cdot B = \begin{bmatrix} 1 & -3 & 2 \\ -3 & 4 & -1 \\ 2 & -1 & 0 \end{bmatrix} \cdot \begin{bmatrix} 3 \\ 4 \\ 6 \end{bmatrix}$$ Perform the matrix multiplication: $$X = \begin{bmatrix} 1 \times 3 + (-3) \times 4 + 2 \times 6 \\ -3 \times 3 + 4 \times 4 + (-1) \times 6 \\ 2 \times 3 + (-1) \times 4 + 0 \times 6 \end{bmatrix}$$ $$X = \begin{bmatrix} 3 - 12 + 12 \\ -9 + 16 - 6 \\ 6 - 4 + 0 \end{bmatrix} = \begin{bmatrix} 3 \\ 1 \\ 2 \end{bmatrix}$$ ### Solution: $$x = 3, \quad y = 1, \quad z = 2$$ Would you like a deeper explanation of any step, or have any questions? Here are five related problems you might consider: 1. Solve a 3x3 system using Gaussian elimination. 2. Find the inverse of a 3x3 matrix without using determinants. 3. Solve a system of equations using Cramer's rule. 4. Solve a system of nonlinear equations using matrix methods. 5. Analyze the solution of a 2x2 matrix system with a singular matrix. **Tip:** When solving systems of equations, checking the determinant first can quickly indicate whether the matrix is invertible.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Matrix Inversion
Formulas
A · X = B (Matrix Equation)
X = A^{-1} · B (Solution using Inverse Matrix)
Determinant of 3x3 Matrix
Cofactor and Adjugate Matrix
Theorems
Matrix Inversion Theorem
Determinant Theorem (invertibility condition: det(A) ≠ 0)
Suitable Grade Level
Grades 10-12, Undergraduate
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