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The question asks whether the inverse of matrix \( A \) is correctly given as:

\[
A = \begin{bmatrix} 
3 & 2 & -1 \\
1 & 6 & 3 \\
2 & -4 & 0 
\end{bmatrix}
\]
and its inverse \( A^{-1} \) as:

\[
A^{-1} = \frac{1}{64} \begin{bmatrix} 
12 & 4 & 12 \\
6 & 2 & -10 \\
-16 & 16 & 16 
\end{bmatrix}
\]

To check the correctness of the inverse matrix, we will:

1. Verify if the determinant of matrix \( A \) is non-zero (since only matrices with non-zero determinant have an inverse).
2. Check if the provided inverse satisfies the property \( A \times A^{-1} = I \), where \( I \) is the identity matrix.

Let me compute the determinant and confirm if the inverse is correct.The determinant of matrix \( A \) is 64, which means that matrix \( A \) is invertible (since the determinant is non-zero). Now, I will check if the given matrix is indeed the inverse by multiplying \( A \) with the proposed inverse matrix and verifying if the result is the identity matrix.The product of \( A \) and the given inverse matrix results in the identity matrix:

\[
\begin{bmatrix} 
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 
\end{bmatrix}
\]

This confirms that the provided inverse is correct. Therefore, the answer to the question is **"صواب" (Correct)**.

Would you like any further details?

Here are five related questions to explore further:
1. How is the determinant of a 3x3 matrix calculated?
2. What are the conditions for a matrix to have an inverse?
3. How do you compute the inverse of a matrix manually?
4. What are some properties of the identity matrix?
5. Why is the determinant important in linear algebra?

**Tip:** Always verify the determinant of a matrix before attempting to compute its inverse to ensure the matrix is invertible.

The inverse of matrix A is provided. The question asks if the given matrix inverse is correct or incorrect: اجب بصح أو خط

In this problem, we verify the correctness of the inverse of matrix A by calculating the determinant and performing matrix multiplication. Learn how to compute the inverse of a 3x3 matrix and check its validity step-by-step. Suitable for students in Grades 11-12 or university-level courses.

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