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We are given matrix \( A = \begin{bmatrix} 1 & 2 & -1 \\ -1 & 1 & 2 \\ 2 & -1 & 1 \end{bmatrix} \), and asked to compute \( \log_{\det(A)} \left( \det \left( \text{adj} \left( \text{adj}(A) \right) \right) \right) \), where \( \det(A) \) is the determinant of matrix \( A \), and \( \text{adj}(A) \) is the adjugate of matrix \( A \).

Let's break this down step by step:

### 1. Find the determinant of \( A \):
First, we compute the determinant of matrix \( A \).

\[
\det(A) = \text{det} \begin{bmatrix} 1 & 2 & -1 \\ -1 & 1 & 2 \\ 2 & -1 & 1 \end{bmatrix}
\]

I will calculate the determinant.The determinant of matrix \( A \) is \( \det(A) = 14 \).

### 2. Adjugate and its properties:
The adjugate of a matrix \( A \), denoted as \( \text{adj}(A) \), has the property:
\[
\det(\text{adj}(A)) = (\det(A))^{n-1}
\]
for an \( n \times n \) matrix, where \( n \) is the order of the matrix. In this case, since \( A \) is a \( 3 \times 3 \) matrix, we have:
\[
\det(\text{adj}(A)) = (\det(A))^2
\]
So, for \( A \), this becomes:
\[
\det(\text{adj}(A)) = (14)^2 = 196
\]

### 3. Apply the adjugate again:
Next, we need to find \( \det(\text{adj}(\text{adj}(A))) \). Using the same formula, we can apply the adjugate property again:
\[
\det(\text{adj}(\text{adj}(A))) = (\det(\text{adj}(A)))^2
\]
Substituting the value of \( \det(\text{adj}(A)) = 196 \):
\[
\det(\text{adj}(\text{adj}(A))) = 196^2
\]

Let me calculate \( 196^2 \).It seems like I can’t do more advanced data analysis right now. Please try again later.

However, I can guide you through this step manually. You can compute the value of \( 196^2 \) directly, which equals:

\[
196^2 = 38416
\]

### 4. Calculate the logarithm:
Now, we need to compute the logarithm base \( \det(A) = 14 \) of \( \det(\text{adj}(\text{adj}(A))) = 38416 \):
\[
\log_{14}(38416)
\]
Since \( 14^4 = 38416 \), we have:
\[
\log_{14}(38416) = 4
\]

Thus, the final answer is \( \log_{14} \left( \det(\text{adj}(\text{adj}(A))) \right) = 4 \).

Would you like further details or have any questions?

Here are 5 related questions to deepen your understanding:
1. How do you compute the adjugate of a matrix step by step?
2. What is the significance of the adjugate in linear algebra?
3. How do you calculate the determinant of larger matrices efficiently?
4. How does the determinant change when applying transformations to a matrix?
5. What are other properties of determinants that can simplify matrix operations?

**Tip**: The adjugate matrix plays a crucial role in calculating the inverse of a matrix for non-singular matrices.

If matrix A = [[1, 2, -1], [-1, 1, 2], [2, -1, 1]], then log_base_detA(det(adj(adj(A)))) is equal to what?

Solve the logarithmic determinant problem involving matrix A = [[1, 2, -1], [-1, 1, 2], [2, -1, 1]], finding the value of log_base_detA(det(adj(adj(A)))) step-by-step. Covers determinant and adjugate matrix properties.

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