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

To solve the system of equations using matrix inversion, we first need to express the system in matrix form \( AX = B \).

Given the system of equations:

\[
\begin{align*}
2x - y - z &= 7 \quad \text{(Equation 1)} \\
3x + y - z &= 7 \quad \text{(Equation 2)} \\
x + y - z &= 3 \quad \text{(Equation 3)}
\end{align*}
\]

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.

Learn how to solve a system of linear equations using matrix inversion. This detailed solution involves expressing equations in matrix form, calculating the inverse of the coefficient matrix, and finding the solution matrix. Suitable for advanced high school and college-level students.

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