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The problem asks to determine for which values of \(\beta\) the matrix \(A\) has an inverse. The matrix given is:

\[
A = \begin{pmatrix} 1 & 1 & 1 \\ \beta & 0 & 2 \\ 2 & -1 & 0 \end{pmatrix}
\]

To find when the matrix is invertible, we need to determine when the determinant of the matrix is non-zero. A matrix is invertible if and only if its determinant is not zero.

### Step-by-Step Solution:

The determinant of a 3x3 matrix \(A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}\) is calculated as:

\[
\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
\]

For the matrix \(A = \begin{pmatrix} 1 & 1 & 1 \\ \beta & 0 & 2 \\ 2 & -1 & 0 \end{pmatrix}\), let's compute the determinant:

1. **First row:**
   - \(a = 1\), \(b = 1\), \(c = 1\)
   
2. **Submatrices for cofactor expansion:**
   - For \(a = 1\): \(\text{det}\begin{pmatrix} 0 & 2 \\ -1 & 0 \end{pmatrix} = (0)(0) - (2)(-1) = 2\)
   - For \(b = 1\): \(\text{det}\begin{pmatrix} \beta & 2 \\ 2 & 0 \end{pmatrix} = (\beta)(0) - (2)(2) = -4\)
   - For \(c = 1\): \(\text{det}\begin{pmatrix} \beta & 0 \\ 2 & -1 \end{pmatrix} = (\beta)(-1) - (0)(2) = -\beta\)

3. **Final determinant calculation:**
   \[
   \text{det}(A) = 1(2) - 1(-4) + 1(-\beta) = 2 + 4 - \beta = 6 - \beta
   \]

### Conclusion:

The matrix \(A\) is invertible when the determinant is non-zero, so:

\[
6 - \beta \neq 0 \quad \Rightarrow \quad \beta \neq 6
\]

Thus, the matrix \(A\) has an inverse for all values of \(\beta\) except \(\beta = 6\).

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Would you like more details on matrix inversion or any other concepts? Here are some related questions for practice:

1. How do you calculate the inverse of a 3x3 matrix?
2. Why is the determinant important for matrix invertibility?
3. What happens if \(\beta = 6\) for the matrix \(A\)?
4. How can matrix inversion be applied in solving systems of linear equations?
5. What is the geometric interpretation of a matrix having no inverse?

**Tip:** Always check the determinant before attempting to find the inverse of a matrix.

Estudia para qué valores de β existe la inversa de la siguiente matriz A = [[1, 1, 1], [β, 0, 2], [2, -1, 0]]

Find out for which values of β the matrix A = [[1, 1, 1], [β, 0, 2], [2, -1, 0]] is invertible. Learn the determinant formula and key concepts in linear algebra to solve this problem. Suitable for high school and university students.

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