https://file.aiquickdraw.com/mathbot/file/17269138459680vauqkg3.jpg

To help you solve the matrix problem based on the image you've uploaded, I'll walk through the visible steps and concepts used.

### Steps to solve the matrix:

1. **Matrix Setup**:  
   The image displays a system of linear equations in matrix form. The equations are written as:
   
   \[
   2x_1 + 8x_2 + 8x_3 = 0
   \]
   \[
   8x_1 + 2x_2 + 2x_3 = 0
   \]
   \[
   8x_1 + 2x_2 + 2x_3 = 0
   \]
   
   These equations are represented by a coefficient matrix \( A \), and it seems the equations have been written as a matrix equation \( A \cdot \mathbf{x} = 0 \).

2. **Matrix Representation**:
   The coefficient matrix \( A \) can be derived from the equations:
   \[
   A = \begin{bmatrix} 
   2 & 8 & 8 \\ 
   8 & 2 & 2 \\ 
   8 & 2 & 2
   \end{bmatrix}
   \]
   This is multiplied by the variable vector \( \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \), which leads to the system of linear equations.

3. **Matrix Determinant**:  
   In order to determine whether the system has non-trivial solutions, we need to find the determinant of matrix \( A \). If the determinant \( \det(A) = 0 \), the system has infinitely many solutions (dependent system). Otherwise, it has only the trivial solution (independent system).

4. **Finding the Determinant**:
   The determinant of a 3x3 matrix \( A \) is calculated as follows:

   \[
   \det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
   \]
   Where:
   \[
   A = \begin{bmatrix}
   a & b & c \\
   d & e & f \\
   g & h & i
   \end{bmatrix}
   \]

   For matrix \( A = \begin{bmatrix} 2 & 8 & 8 \\ 8 & 2 & 2 \\ 8 & 2 & 2 \end{bmatrix} \), the determinant calculation would look like:

   \[
   \det(A) = 2((2 \times 2) - (2 \times 2)) - 8((8 \times 2) - (2 \times 8)) + 8((8 \times 2) - (8 \times 2))
   \]

   Let me calculate this determinant explicitly for you.The determinant of the matrix \( A \) is 0, meaning that the system of equations is **dependent**. This indicates that there are infinitely many solutions to the system, rather than just the trivial solution.

### Next Steps:
1. **Solve for General Solutions**: Since the determinant is 0, you can either use row reduction (Gaussian elimination) or other methods like the adjugate matrix method to find the general solution for this system.

2. **Rank of the Matrix**: The rank of the matrix is less than 3 (because the determinant is 0), so you will need to explore the relationships between the equations to express one variable in terms of others.

Do you want further details on the row reduction process or other techniques to find the solution?

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Here are 5 related questions to expand your understanding:
1. What does it mean for a system of equations to be dependent or independent?
2. How can Gaussian elimination help in finding solutions to a matrix equation?
3. How do you find the rank of a matrix using row reduction?
4. What are the implications of a zero determinant for solving linear equations?
5. How can we interpret the null space of a matrix in this context?

**Tip**: When solving systems of equations with infinite solutions, you often express one or more variables in terms of free variables.

Learn how to solve a 3x3 matrix system of linear equations using determinant calculation and understand why this system has infinitely many solutions. This problem involves key linear algebra concepts, including matrix determinants and dependent systems.

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