if |A| = 2 and adj A = [[- 2, 3, 1], [6, - 8, - 2], [- 4, 7, 1]] then find A.

Given that \(|A| = 2\) and the adjugate matrix \(\text{adj}(A)\) is:

\[
\text{adj}(A) = \begin{bmatrix} -2 & 3 & 1 \\ 6 & -8 & -2 \\ -4 & 7 & 1 \end{bmatrix}
\]

We need to find the matrix \(A\).

### Key Formula
We use the relationship between a matrix \(A\), its determinant \(|A|\), and its adjugate \(\text{adj}(A)\):

\[
A = \frac{1}{|A|} \cdot \text{adj}(A)
\]

### Calculation
Substituting the given values:

\[
A = \frac{1}{2} \cdot \begin{bmatrix} -2 & 3 & 1 \\ 6 & -8 & -2 \\ -4 & 7 & 1 \end{bmatrix}
\]

\[
A = \begin{bmatrix} \frac{-2}{2} & \frac{3}{2} & \frac{1}{2} \\ \frac{6}{2} & \frac{-8}{2} & \frac{-2}{2} \\ \frac{-4}{2} & \frac{7}{2} & \frac{1}{2} \end{bmatrix}
\]

Simplifying:

\[
A = \begin{bmatrix} -1 & \frac{3}{2} & \frac{1}{2} \\ 3 & -4 & -1 \\ -2 & \frac{7}{2} & \frac{1}{2} \end{bmatrix}
\]

### Final Answer
The matrix \(A\) is:

\[
A = \begin{bmatrix} -1 & \frac{3}{2} & \frac{1}{2} \\ 3 & -4 & -1 \\ -2 & \frac{7}{2} & \frac{1}{2} \end{bmatrix}
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. What is the adjugate matrix of a \(3 \times 3\) matrix?
2. How do you calculate the determinant of a matrix?
3. What is the inverse of a matrix?
4. How does the adjugate relate to the inverse matrix?
5. Can a matrix with a zero determinant have an adjugate?

**Tip:** The adjugate matrix is the transpose of the cofactor matrix.

Discover how to find matrix A when given its determinant |A| = 2 and adjugate matrix adj(A). This problem involves advanced matrix algebra concepts and their applications, including the formula for matrix inverse. Suitable for advanced level students and professionals.

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