Find the maximum value of the determinant of an arbitrary `3xx3` matrix A, each of whose entries `a_(ij) in {-1,1}`.

Answer

To find the maximum value of the determinant of a \(3 \times 3\) matrix \(A\) where each entry \(a_{ij}\) is either \(-1\) or \(1\), we can follow these steps:

### Step 1: Define the Matrix
Let the matrix \(A\) be defined as follows:
\[ A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} \]
where \(a_{ij} \in \{-1, 1\}\).

### Step 2: Calculate the Determinant
The determinant of the matrix \(A\) can be calculated using the formula for the determinant of a \(3 \times 3\) matrix:
\[ \text{det}(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31}) \]

### Step 3: Analyze the Terms
Each term in the determinant expression consists of products of the entries of the matrix. Since each entry can only be \(-1\) or \(1\), the products will also yield values of either \(-1\) or \(1\).

### Step 4: Consider Combinations
To maximize the determinant, we need to consider combinations of the entries of the matrix. The maximum determinant can be achieved when the matrix is structured to maximize the positive contributions and minimize the negative contributions.

### Step 5: Test Possible Matrices
One effective approach is to test matrices where all entries are \(1\) or where they alternate between \(1\) and \(-1\). For example, consider the following matrix:
\[ A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & -1 \\ 1 & -1 & 1 \end{bmatrix} \]
Calculating the determinant:
\[ \text{det}(A) = 1(1 \cdot 1 - (-1) \cdot (-1)) - 1(1 \cdot 1 - (-1) \cdot 1) + 1(1 \cdot (-1) - 1 \cdot 1) \]
\[ = 1(1 - 1) - 1(1 - (-1)) + 1(-1 - 1) \]
\[ = 0 - 1(2) - 2 = -4 \]

### Step 6: Find Maximum Determinant
After testing various combinations, we find that the maximum determinant occurs with the following matrix:
\[ A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & -1 & -1 \\ 1 & -1 & 1 \end{bmatrix} \]
Calculating the determinant:
\[ \text{det}(A) = 1((-1)(1) - (-1)(-1)) - 1(1(1) - (-1)(1)) + 1(1(-1) - 1(1)) \]
\[ = 1(-1 - 1) - 1(1 + 1) + 1(-1 - 1) \]
\[ = 1(-2) - 1(2) + 1(-2) = -2 - 2 - 2 = -6 \]

### Conclusion
The maximum value of the determinant of the matrix \(A\) with entries in \(\{-1, 1\}\) is \(6\).