https://file.aiquickdraw.com/mathbot/file/1727107644815ldyuey74.jpg

The question asks to find the inverse of the following 3x3 matrix:

\[
\begin{pmatrix}
1 & 1 & 1 \\
2 & -1 & 1 \\
3 & -2 & -1
\end{pmatrix}
\]

To find the inverse of a 3x3 matrix, we can use the formula involving the adjugate and the determinant of the matrix. The inverse of a matrix \( A \) is given by:

\[
A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A)
\]

Where:
- \(\text{det}(A)\) is the determinant of the matrix.
- \(\text{adj}(A)\) is the adjugate matrix, which is the transpose of the cofactor matrix.

Let me calculate the determinant of the matrix and, if it's non-zero, find its inverse.The determinant of the matrix is 7, which is non-zero. This means the matrix is invertible. Now, I'll compute the inverse of the matrix.The inverse of the matrix is:

\[
A^{-1} = \begin{pmatrix}
0.4286 & -0.1429 & 0.2857 \\
0.7143 & -0.5714 & 0.1429 \\
-0.1429 & 0.7143 & -0.4286
\end{pmatrix}
\]

Let me know if you'd like further details or explanations!

Here are five questions to deepen your understanding:
1. What does it mean for a matrix to be invertible?
2. How do you interpret the determinant in the context of matrix inversion?
3. Can a matrix with a zero determinant have an inverse? Why or why not?
4. How is the cofactor matrix related to finding the inverse of a matrix?
5. Why is it necessary to transpose the cofactor matrix when calculating the adjugate?

**Tip:** Always check the determinant first when determining if a matrix has an inverse; a determinant of zero means the matrix is not invertible!

Dapatkan invers matrik berikut: [1, 1, 1; 2, -1, 1; 3, -2, -1]

Learn how to find the inverse of a 3x3 matrix using the determinant and adjugate matrix method. Detailed step-by-step solution for the matrix [1, 1, 1; 2, -1, 1; 3, -2, -1]. Suitable for students in Grades 10-12.

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