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The matrix \( A \) given is:

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

We need to find the inverse matrix \( A^{-1} \) and determine the values for \( a, b, c, d, e, f, g, h, i \).

To calculate the inverse of a 3x3 matrix, we use the following steps:
1. Compute the determinant of the matrix \( A \).
2. Find the matrix of minors.
3. Form the matrix of cofactors.
4. Compute the adjugate (transpose of the cofactor matrix).
5. Divide each term of the adjugate by the determinant of \( A \).

I'll calculate the inverse and fill in the blanks for you.The inverse of the matrix \( A \) is:

\[
A^{-1} = \begin{pmatrix}
-1 & 2 & -4 \\
1  & -1 & 3 \\
0  & 0  & 1
\end{pmatrix}
\]

Thus, the values for \( a, b, c, d, e, f, g, h, i \) are:

\[
a = -1, \quad b = 2, \quad c = -4
\]
\[
d = 1, \quad e = -1, \quad f = 3
\]
\[
g = 0, \quad h = 0, \quad i = 1
\]

Would you like more details or have any other questions? Here are 5 related questions to deepen your understanding:

1. How do you calculate the determinant of a 3x3 matrix?
2. What is the matrix of minors and how do you find it?
3. How does the cofactor matrix relate to the matrix of minors?
4. What is the significance of the adjugate matrix in finding the inverse?
5. How does the inverse matrix help in solving systems of linear equations?

**Tip:** The determinant of a matrix is crucial in determining whether a matrix is invertible. If the determinant is zero, the matrix has no inverse.

Let A = [[1, 2, -2], [1, 1, 1], [0, 0, 1]]. Fill in the blanks for the inverse matrix A^-1 = [[a, b, c], [d, e, f], [g, h, i]].

Learn how to calculate the inverse of a 3x3 matrix with a step-by-step breakdown, including determinant calculation, cofactor matrix formation, and adjugate matrix usage. This linear algebra problem is perfect for college-level students.

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