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 start by expressing the system in matrix form as \( AX = B \), where:

\[
A = \begin{pmatrix}
2 & -1 & -1 \\
3 & 1 & -1 \\
1 & 1 & -1
\end{pmatrix},
\quad
X = \begin{pmatrix}
x \\
y \\
z
\end{pmatrix},
\quad
B = \begin{pmatrix}
7 \\
7 \\
3
\end{pmatrix}
\]

The solution \( X \) is given by \( X = A^{-1}B \), where \( A^{-1} \) is the inverse of matrix \( A \).

### Step 1: Find the determinant of \( A \)

The determinant of matrix \( A \) is calculated as follows:

\[
\text{det}(A) = 2\left(1(-1) - 1(-1)\right) - (-1)\left(3(-1) - 1(-1)\right) + (-1)\left(3(1) - 1(1)\right)
\]

Simplifying each part:

\[
\text{det}(A) = 2(0) - (-1)(-4) + (-1)(2) = 0 - 4 - 2 = -6
\]

### Step 2: Find the inverse of \( A \)

Given \( A \) is invertible (since \( \text{det}(A) \neq 0 \)), the inverse \( A^{-1} \) can be found using the formula:

\[
A^{-1} = \frac{1}{\text{det}(A)} \text{adj}(A)
\]

Where \( \text{adj}(A) \) is the adjugate (transpose of the cofactor matrix) of \( A \). After calculating the cofactors, the adjugate matrix is:

\[
\text{adj}(A) = \begin{pmatrix}
-1 & 2 & 2 \\
-1 & 1 & 5 \\
-4 & 4 & 7
\end{pmatrix}
\]

So,

\[
A^{-1} = \frac{1}{-6} \begin{pmatrix}
-1 & 2 & 2 \\
-1 & 1 & 5 \\
-4 & 4 & 7
\end{pmatrix} = \begin{pmatrix}
\frac{1}{6} & -\frac{1}{3} & -\frac{1}{3} \\
\frac{1}{6} & -\frac{1}{6} & -\frac{5}{6} \\
\frac{2}{3} & -\frac{2}{3} & -\frac{7}{6}
\end{pmatrix}
\]

### Step 3: Multiply \( A^{-1} \) by \( B \)

Now, multiply \( A^{-1} \) by \( B \):

\[
X = \begin{pmatrix}
\frac{1}{6} & -\frac{1}{3} & -\frac{1}{3} \\
\frac{1}{6} & -\frac{1}{6} & -\frac{5}{6} \\
\frac{2}{3} & -\frac{2}{3} & -\frac{7}{6}
\end{pmatrix}
\begin{pmatrix}
7 \\
7 \\
3
\end{pmatrix}
\]

This results in:

\[
X = \begin{pmatrix}
1 \\
2 \\
3
\end{pmatrix}
\]

### Solution:
The solution to the system of equations is:

\[
x = 1, \quad y = 2, \quad z = 3
\]

Would you like details on any particular step or further explanation?

Here are some related questions:

1. How is the determinant of a matrix calculated in general?
2. What is the significance of the determinant being non-zero?
3. Can you show how to calculate the adjugate matrix in more detail?
4. How does matrix inversion relate to solving linear systems?
5. What are some cases where matrix inversion may not work?

**Tip:** When solving a system of equations using matrix inversion, always check if the determinant is non-zero. A zero determinant means the matrix is singular and non-invertible, indicating either no solutions or infinitely many solutions.

Learn how to solve a system of linear equations using matrix inversion with a step-by-step explanation. Understand matrix operations and the importance of determinant in matrix inversion. Suitable for advanced high school students.

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