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You have provided a 4x4 matrix \( A \):

\[
A = \begin{pmatrix}
1 & -1 & 0 & 2 \\
0 & 1 & 1 & -1 \\
2 & 1 & 2 & 1 \\
3 & 2 & 1 & 6
\end{pmatrix}
\]

I will calculate the inverse of this matrix \( A \). Let me perform the necessary steps to find it.The inverse of the given matrix \( A \) is:

\[
A^{-1} = \begin{pmatrix}
-2 & -3.4 & 1.8 & -0.2 \\
-1 & -0.6 & 0.2 & 0.2 \\
2 & 3 & -1 & 0 \\
1 & 1.4 & -0.8 & 0.2
\end{pmatrix}
\]

Let me know if you need any further clarification or details!

Here are some related questions to deepen your understanding:

1. How can you verify if a matrix is invertible?
2. What is the significance of the determinant in calculating the inverse of a matrix?
3. Can every square matrix be inverted? Why or why not?
4. How does matrix inversion apply to solving systems of linear equations?
5. How can the properties of determinants help in finding the inverse more efficiently?

**Tip:** When dealing with larger matrices, it’s easier to use computational tools (like Python or a graphing calculator) to find the inverse.

Find the inverse of the 4x4 matrix A = [[1, -1, 0, 2], [0, 1, 1, -1], [2, 1, 2, 1], [3, 2, 1, 6]].

Learn how to find the inverse of a 4x4 matrix A = [[1, -1, 0, 2], [0, 1, 1, -1], [2, 1, 2, 1], [3, 2, 1, 6]]. This problem covers key linear algebra concepts including matrix inversion, the Invertible Matrix Theorem, and the calculation of determinants and cofactors. Suitable for advanced high school or undergraduate students.

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