Math Problem Statement
solve by matrix inversion: 2x-y-z=7, 3x+y-z=7, x+y-z=3
Solution
To solve the system of equations using matrix inversion, we first need to express the system in matrix form .
Given the system of equations:
2x - y - z &= 7 \quad \text{(Equation 1)} \\ 3x + y - z &= 7 \quad \text{(Equation 2)} \\ x + y - z &= 3 \quad \text{(Equation 3)} \end{align*}$$ This can be written in matrix form as: $$\begin{pmatrix} 2 & -1 & -1 \\ 3 & 1 & -1 \\ 1 & 1 & -1 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 7 \\ 7 \\ 3 \end{pmatrix}$$ Here, - Matrix $$A$$ is the coefficient matrix: $$A = \begin{pmatrix} 2 & -1 & -1 \\ 3 & 1 & -1 \\ 1 & 1 & -1 \end{pmatrix}$$ - Vector $$X$$ is the variable matrix: $$X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$$ - Vector $$B$$ is the constant matrix: $$B = \begin{pmatrix} 7 \\ 7 \\ 3 \end{pmatrix}$$ The solution can be found by calculating $$X = A^{-1}B$$, where $$A^{-1}$$ is the inverse of matrix $$A$$. Let's calculate the inverse of matrix $$A$$ and then multiply it by matrix $$B$$ to find $$X$$.The solution to the system of equations is: $$x = 2, \quad y = -1, \quad z = -2$$ This means the values of $$x$$, $$y$$, and $$z$$ that satisfy all three equations are $$x = 2$$, $$y = -1$$, and $$z = -2$$. Would you like further details or explanations on this process? Here are 5 related questions to expand your understanding: 1. How do you calculate the determinant of a 3x3 matrix? 2. What are the conditions under which a matrix has an inverse? 3. How would you solve the system if the matrix $$A$$ were not invertible? 4. Can matrix inversion be applied to systems with more or fewer than three variables? 5. What are some alternative methods to solve systems of linear equations? **Tip:** When dealing with systems of linear equations, always check if the determinant of the coefficient matrix is non-zero. A zero determinant indicates that the matrix is singular and does not have an inverse, meaning the system may have no solution or infinitely many solutions.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Matrix Algebra
Linear Equations
Formulas
Matrix inversion formula: X = A^-1 * B
Theorems
Invertibility of matrices
Suitable Grade Level
Advanced High School / College Level