https://r2.erweima.ai/mathbot/image/1725503964548r96tpgs2.jpg

The image contains three mathematical problems. Here's a breakdown of each problem:

### 1. System of Linear Equations
- **Problem**: Solve the following system of equations:
  \[
  \begin{cases}
  3x + y + 2z = 0 \\
  2x + y - z = 0 \\
  x + 2y + 2z = 0
  \end{cases}
  \]

### 2. Matrix Equation Problem
- **Problem**: Determine the real values of the parameter \(m \in \mathbb{R}\) for which the following system has a unique solution:
  \[
  \begin{cases}
  x_1 + x_2 + mx_3 = 1 \\
  x_1 + x_3 = 1 \\
  x_2 + mx_3 = 1
  \end{cases}
  \]

### 3. Matrix Determinant Problem
- **Problem**: Solve the following matrix determinant problem:
  \[
  \text{Let } A = 
  \begin{pmatrix}
  a & 1 \\
  1 & 0
  \end{pmatrix}
  \text{ and find the values of } a \in \mathbb{R} \text{ such that } A^2 = I_2, \text{ where } I_2 \text{ is the identity matrix.}
  \]
  The given options are:
  - (a) \(a = 0\)
  - (b) \(a = 1\)
  - (c) \(a = \sqrt{2}\)
  - (d) \(a = -2\)

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Solve the following system of equations and matrix determinant problem.

Learn how to solve a system of linear equations and a matrix determinant problem. This example covers key concepts in linear algebra, including system of equations and matrix identities, suitable for high school students.

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