https://file.aiquickdraw.com/mathbot/file/1727722334916srvd75jk.jpg

From the image, we are given a system of linear equations:

\[
\begin{aligned}
x + 2y + 2z &= 1 \\
-2x - y + z &= 1 \\
2x + 2y + 2z &= 4
\end{aligned}
\]

1. **Coefficient matrix (A)** of the system:
\[
A = \begin{bmatrix}
1 & 2 & 2 \\
-2 & -1 & 1 \\
2 & 2 & 2
\end{bmatrix}
\]

2. **Task:** Compute the inverse of the matrix \( A \).

Let me compute the inverse matrix for you.The inverse of the matrix \( A \) is:

\[
A^{-1} = \begin{bmatrix}
-1 & 0 & 1 \\
1.5 & -0.5 & -1.25 \\
-0.5 & 0.5 & 0.75
\end{bmatrix}
\]

### Now, let's address the statement about whether this system has a unique solution:

- A system of linear equations has a unique solution if the coefficient matrix \( A \) is invertible (i.e., if the determinant of \( A \neq 0 \)).
- Since the inverse of \( A \) exists, the determinant is non-zero, meaning the system **does** have a unique solution.

Would you like more details on this, or do you have any other questions?

Here are some related questions for further exploration:

1. How is the determinant of a matrix calculated, and what does it signify?
2. What are the conditions for a matrix to have an inverse?
3. How can you verify the solution of a system of equations using matrix inversion?
4. What does it mean if the determinant of a matrix is zero?
5. How can we check if a system of equations is consistent?

**Tip:** When solving systems of equations, always check if the matrix is invertible to determine if there is a unique solution.

What is the inverse of the matrix, and does the system of linear equations have a unique solution?

This problem focuses on finding the inverse of a 3x3 matrix and determining if a system of linear equations has a unique solution. The matrix is invertible, indicating a unique solution exists. Learn about matrix inversion, determinants, and how they relate to system solvability.

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