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(A A^T) =3`. Where `tr(A)` is the trace of A.

Options

A. 572
B. 612
C. 672
D. 682

Explanation

To solve the problem, we need to find the number of \(3 \times 3\) matrices \(A\) with elements from the set \(\{-1, 0, 1\}\) such that the trace of \(AA^T\) equals 3.

### Step-by-Step Solution:

1. **Understanding the Trace of \(AA^T\)**:
The trace of \(AA^T\) is equal to the sum of the squares of all the elements of the matrix \(A\). If \(A\) is a \(3 \times 3\) matrix, we can denote its elements 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 \(tr(AA^T)\) is given by:
\[ tr(AA^T) = a_{11}^2 + a_{12}^2 + a_{13}^2 + a_{21}^2 + a_{22}^2 + a_{23}^2 + a_{31}^2 + a_{32}^2 + a_{33}^2 \]

2. **Setting Up the Equation**:
We need to find the number of matrices such that:
\[ a_{11}^2 + a_{12}^2 + a_{13}^2 + a_{21}^2 + a_{22}^2 + a_{23}^2 + a_{31}^2 + a_{32}^2 + a_{33}^2 = 3 \]
Since each element \(a_{ij}\) can take values from \(\{-1, 0, 1\}\), we have:
- \(a_{ij}^2 = 1\) if \(a_{ij} = 1\) or \(a_{ij} = -1\)
- \(a_{ij}^2 = 0\) if \(a_{ij} = 0\)

3. **Counting Non-Zero Entries**:
For the sum of squares to equal 3, exactly 3 entries of the matrix \(A\) must be either \(1\) or \(-1\) (since \(1^2 = 1\) and \((-1)^2 = 1\)). The remaining 6 entries must be \(0\).

4. **Choosing Positions for Non-Zero Entries**:
We need to choose 3 positions from the 9 available in the matrix for the non-zero entries. The number of ways to choose 3 positions from 9 is given by the combination:
\[ \binom{9}{3} \]

5. **Assigning Values to Non-Zero Entries**:
Each of the 3 chosen positions can independently be either \(1\) or \(-1\). Thus, for each of the 3 positions, we have 2 choices (either \(1\) or \(-1\)). Therefore, the total number of ways to assign values to these 3 positions is:
\[ 2^3 = 8 \]

6. **Calculating the Total Number of Matrices**:
The total number of matrices \(A\) that satisfy the condition \(tr(AA^T) = 3\) is given by multiplying the number of ways to choose the positions by the number of ways to assign values:
\[ \text{Total Matrices} = \binom{9}{3} \times 2^3 \]

7. **Calculating \(\binom{9}{3}\)**:
\[ \binom{9}{3} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84 \]

8. **Final Calculation**:
\[ \text{Total Matrices} = 84 \times 8 = 672 \]

### Conclusion:
The number of \(3 \times 3\) matrices \(A\) such that \(tr(AA^T) = 3\) is **672**.