Math Problem Statement

solve by matrix inversion: 2x - y - z = 7; 3x + y - z = 7; x + y - z = 3

Solution

We are given the system of linear equations:

2x - y - z &= 7 \quad (1) \\ 3x + y - z &= 7 \quad (2) \\ x + y - z &= 3 \quad (3) \end{aligned}$$ To solve this using matrix inversion, we can express the system in matrix form as $$AX = B$$, where: $$A = \begin{bmatrix} 2 & -1 & -1 \\ 3 & 1 & -1 \\ 1 & 1 & -1 \end{bmatrix}, \quad X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \quad B = \begin{bmatrix} 7 \\ 7 \\ 3 \end{bmatrix}$$ To find $$X$$, we calculate: $$X = A^{-1}B$$ Let's first compute $$A^{-1}$$, the inverse of matrix $$A$$, and then multiply it by $$B$$ to find $$X$$.The solution to the system of equations is: $$x = 2, \quad y = -1, \quad z = -2$$ Would you like further details or have any questions? Here are five related questions: 1. How do you determine if a matrix is invertible? 2. What happens if a matrix is not invertible in such problems? 3. Can matrix inversion be used for larger systems, such as a 4x4 system? 4. What other methods can be used to solve systems of linear equations? 5. How is the determinant of a matrix related to its invertibility? **Tip:** Always check the determinant of a matrix before attempting inversion. If the determinant is zero, the matrix is not invertible.

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Math Problem Analysis

Mathematical Concepts

Linear equations
Matrix inversion

Formulas

Matrix inversion: X = A^{-1}B

Theorems

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Suitable Grade Level

Grades 10-12