To solve the problem of finding how many matrices \( X \) with entries from the set \{0, 1, 2\} exist such that the sum of the diagonal entries of \( X \cdot X^T \) equals 7, we can follow these steps:
### Step 1: Define the Matrix
Let \( X \) be a \( 3 \times 3 \) matrix with entries \( a, b, c, d, e, f, g, h, i \). Thus, we can represent \( X \) as:
\[
X = \begin{pmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{pmatrix}
\]
### Step 2: Calculate \( X \cdot X^T \)
The product \( X \cdot X^T \) will yield a \( 3 \times 3 \) matrix. The diagonal entries of this product are given by:
- First diagonal entry: \( a^2 + b^2 + c^2 \)
- Second diagonal entry: \( d^2 + e^2 + f^2 \)
- Third diagonal entry: \( g^2 + h^2 + i^2 \)
Thus, the sum of the diagonal entries of \( X \cdot X^T \) is:
\[
S = a^2 + b^2 + c^2 + d^2 + e^2 + f^2 + g^2 + h^2 + i^2
\]
### Step 3: Set Up the Equation
We need to find the number of matrices \( X \) such that:
\[
S = 7
\]
Given that each entry of \( X \) can be 0, 1, or 2, the possible values for \( x^2 \) (where \( x \) is an entry of \( X \)) are:
- If \( x = 0 \), then \( x^2 = 0 \)
- If \( x = 1 \), then \( x^2 = 1 \)
- If \( x = 2 \), then \( x^2 = 4 \)
### Step 4: Consider Possible Combinations
We need to find combinations of \( a^2, b^2, c^2, d^2, e^2, f^2, g^2, h^2, i^2 \) that sum to 7.
#### Case 1: Seven entries are 1 and the rest are 0
- This means we have 7 entries equal to 1 and 2 entries equal to 0.
- The number of ways to choose 7 positions from 9 is given by:
\[
\binom{9}{7} = \binom{9}{2} = 36
\]
#### Case 2: One entry is 2, three entries are 1, and the rest are 0
- This means we have 1 entry equal to 2, 3 entries equal to 1, and 5 entries equal to 0.
- The number of ways to choose 1 position for 2 from 9 is \( \binom{9}{1} \).
- The number of ways to choose 3 positions for 1 from the remaining 8 is \( \binom{8}{3} \).
- Thus, the total for this case is:
\[
\binom{9}{1} \cdot \binom{8}{3} = 9 \cdot 56 = 504
\]
### Step 5: Total the Cases
Now, we sum the two cases:
\[
\text{Total} = 36 + 504 = 540
\]
### Conclusion
Thus, the total number of matrices \( X \) such that the sum of the diagonal entries of \( X \cdot X^T \) equals 7 is \( \boxed{540} \).
