A is a `3 xx 3` matrix whose elements are from the set `{ -1, 0, 1}`. Find the number of matrices A such that `tr(AxxA^T) =3`. Where `tr(A)` is the trace of A.

Answer

To solve the problem of finding the number of \(3 \times 3\) matrices \(A\) with elements from the set \(\{-1, 0, 1\}\) such that \(\text{tr}(A A^T) = 3\), we can follow these steps:

### Step 1: Understanding the Trace
The trace of a matrix \(A A^T\) is equal to the sum of the squares of its entries. For a \(3 \times 3\) matrix \(A\), the entries can be represented as follows:

\[ A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix} \]

The trace of \(A A^T\) is given by:

\[ \text{tr}(A A^T) = a_{11}^2 + a_{12}^2 + a_{13}^2 + a_{21}^2 + a_{22}^2 + a_{23}^2 + a_{31}^2 + a_{32}^2 + a_{33}^2 \]

### Step 2: Setting Up the Equation
We need to find the number of matrices such that:

\[ \text{tr}(A A^T) = 3 \]

Given that the elements of \(A\) can only be \(-1\), \(0\), or \(1\), we note that:

- \( (-1)^2 = 1 \)
- \( 0^2 = 0 \)
- \( 1^2 = 1 \)

Thus, the only way to achieve a trace of 3 is to have exactly three entries of \(A\) that are either \(1\) or \(-1\) (since each contributes \(1\) to the trace), and the rest must be \(0\).

### Step 3: Counting the Valid Configurations
To achieve a trace of 3, we need to select 3 positions out of the 9 available in the matrix \(A\) to place either \(1\) or \(-1\). The remaining 6 positions will be filled with \(0\).

1. **Choosing Positions**: The number of ways to choose 3 positions from 9 is given by the combination formula:

\[ \binom{9}{3} \]

2. **Choosing Values**: For each of the 3 chosen positions, we can either place \(1\) or \(-1\). This gives us \(2\) choices for each of the 3 positions, leading to:

\[ 2^3 = 8 \]

### Step 4: Final Calculation
The total number of matrices \(A\) satisfying the condition is:

\[ \text{Total} = \binom{9}{3} \times 2^3 \]

Calculating these values:

\[ \binom{9}{3} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84 \]

Thus,

\[ \text{Total} = 84 \times 8 = 672 \]

### Final Answer
The number of \(3 \times 3\) matrices \(A\) such that \(\text{tr}(A A^T) = 3\) is **672**.

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