https://file.aiquickdraw.com/mathbot/file/1727316130269exyna6aq.jpg

The image contains two systems of linear equations that are being solved using inverse matrices. However, the provided solutions are marked as incorrect (indicated by the red "x").

Let's analyze and solve the systems step-by-step using inverse matrices.

### System (a):
\[
\begin{aligned}
x_1 + 2x_2 + x_3 &= 2 \\
x_1 + 2x_2 - x_3 &= 0 \\
x_1 - 2x_2 + x_3 &= -2
\end{aligned}
\]

This system can be written as a matrix equation:

\[
A \cdot X = B
\]

where:
\[
A = \begin{pmatrix} 1 & 2 & 1 \\ 1 & 2 & -1 \\ 1 & -2 & 1 \end{pmatrix}, \quad
X = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}, \quad
B = \begin{pmatrix} 2 \\ 0 \\ -2 \end{pmatrix}
\]

We'll compute the inverse of matrix \( A \) and then multiply it by \( B \) to find \( X \).

### System (b):
\[
\begin{aligned}
x_1 + 2x_2 + x_3 &= 0 \\
x_1 + 2x_2 - x_3 &= -2 \\
x_1 - 2x_2 + x_3 &= 0
\end{aligned}
\]

Similarly, this can also be written as:

\[
A \cdot X = B
\]

where \( A \) remains the same as in system (a), but \( B \) changes:

\[
B = \begin{pmatrix} 0 \\ -2 \\ 0 \end{pmatrix}
\]

Let's solve both systems by finding the inverse of \( A \) and multiplying it by the respective \( B \). I'll calculate this now.Here are the correct solutions for the two systems of linear equations:

### System (a):
\[
x_1 = -1, \quad x_2 = 1, \quad x_3 = 1
\]

### System (b):
\[
x_1 = -1, \quad x_2 = 0, \quad x_3 = 1
\]

Both previously given solutions in the image are incorrect. The correct solutions, obtained using the inverse matrix method, are as shown above.

Do you need further explanation, or do you have any questions?

Here are five related questions to expand your understanding:
1. How do you compute the inverse of a 3x3 matrix manually?
2. Why does using an inverse matrix method work for solving systems of linear equations?
3. What conditions must a matrix satisfy to have an inverse?
4. How can we verify if the solution obtained is correct for both systems?
5. Are there alternative methods to solve a system of linear equations if the inverse matrix method is not applicable?

**Tip:** Always check if the determinant of a matrix is non-zero before attempting to compute its inverse. A zero determinant means the matrix does not have an inverse, and other methods must be used.

Use an inverse matrix to solve each system of linear equations: (a) x1 + 2x2 + x3 = 2, x1 + 2x2 - x3 = 0, x1 - 2x2 + x3 = -2; (b) x1 + 2x2 + x3 = 0, x1 + 2x2 - x3 = -2, x1 - 2x2 + x3 = 0.

Learn how to solve two systems of linear equations using the inverse matrix method. Includes detailed solutions for equations like x1 + 2x2 + x3 = 2, and more. Suitable for high school and college students.

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