https://file.aiquickdraw.com/mathbot/file/1725537069584tp3443m1.jpg

The image asks to find the inverse of a matrix \( A \), where:

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

To find the inverse of matrix \( A \), we can follow these steps:

1. **Find the determinant of \( A \):** This step is important because if the determinant is zero, the matrix is non-invertible.
2. **Calculate the matrix of minors.**
3. **Find the cofactor matrix.**
4. **Transpose the cofactor matrix.**
5. **Multiply by the reciprocal of the determinant to find the inverse.**

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

\[
A^{-1} = \begin{pmatrix}
-0.5 & 1.5 & 0.5 \\
1.5 & -2.5 & -0.5 \\
-0.5 & 0.5 & 0.5
\end{pmatrix}
\]

Would you like a step-by-step breakdown of how the inverse was calculated, or do you have any specific questions?

Here are five related questions to deepen your understanding:
1. How is the determinant of a 3x3 matrix computed?
2. What does it mean if a matrix does not have an inverse?
3. How can you verify if the inverse is correct?
4. What are some applications of matrix inversion in real-world problems?
5. How does finding the inverse of a matrix relate to solving systems of linear equations?

**Tip:** Always check if a matrix is invertible by calculating its determinant first. If the determinant is zero, the matrix has no inverse.

Find the inverse of matrix A, where A = [[2, 1, -1], [1, 0, -1], [1, 1, 2]].

Learn how to find the inverse of the 3x3 matrix A = [[2, 1, -1], [1, 0, -1], [1, 1, 2]] by following detailed steps. This solution covers matrix inversion concepts, determinant calculation, and key linear algebra techniques. Suitable for undergraduate-level mathematics students.

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