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

We are given the system of linear equations:

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

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.

Learn how to solve a system of linear equations using matrix inversion with a step-by-step solution. Understand the concept of matrix A, vector X, and vector B in the context of solving equations. Suitable for students in Grades 10-12.

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