The number of matrices X with entries `{0,2,3}` for which the sum of all the principal diagonal elements of `X.X^(T)` is 28 (where `X^(T)` is the transpose matrix of X), is

Options

A. 12
B. 18
C. 36
D. 44

Answer

To solve the problem, we need to find the number of 3x3 matrices \( X \) with entries from the set \{0, 2, 3\} such that the sum of the principal diagonal elements of \( X \cdot X^T \) equals 28.

### Step-by-step Solution:

1. **Understanding the Matrix Product**:
The matrix \( X \) is a 3x3 matrix. When we compute \( X \cdot X^T \), the resulting matrix will also be a 3x3 matrix. The elements of this matrix are computed as follows:
\[ (X \cdot X^T)_{ij} = \sum_{k=1}^{3} X_{ik} X_{jk} \]
The principal diagonal elements of \( X \cdot X^T \) are given by:
\[ S = (X \cdot X^T)_{11} + (X \cdot X^T)_{22} + (X \cdot X^T)_{33} \]

2. **Calculating the Diagonal Elements**:
The diagonal elements can be calculated as:
- \( (X \cdot X^T)_{11} = a^2 + d^2 + g^2 \)
- \( (X \cdot X^T)_{22} = b^2 + e^2 + h^2 \)
- \( (X \cdot X^T)_{33} = c^2 + f^2 + i^2 \)

Therefore, we have:
\[ S = a^2 + b^2 + c^2 + d^2 + e^2 + f^2 + g^2 + h^2 + i^2 \]

3. **Setting Up the Equation**:
We need to find the combinations of entries such that:
\[ S = 28 \]
The possible squares of the entries from the set \{0, 2, 3\} are:
- \( 0^2 = 0 \)
- \( 2^2 = 4 \)
- \( 3^2 = 9 \)

4. **Finding Combinations**:
Since \( 0^2 \) contributes nothing to the sum, we can only use \( 2 \) and \( 3 \). Let's denote:
- \( x \): number of entries that are \( 2 \)
- \( y \): number of entries that are \( 3 \)

The equation becomes:
\[ 4x + 9y = 28 \]
We also know that \( x + y + z = 9 \) where \( z \) is the number of zeros.

5. **Solving for Non-negative Integer Solutions**:
Rearranging the first equation gives:
\[ 4x = 28 - 9y \quad \Rightarrow \quad x = \frac{28 - 9y}{4} \]
For \( x \) to be a non-negative integer, \( 28 - 9y \) must be non-negative and divisible by 4.

Testing possible values for \( y \):
- If \( y = 0 \): \( 4x = 28 \) → \( x = 7 \)
- If \( y = 1 \): \( 4x = 19 \) → not valid
- If \( y = 2 \): \( 4x = 10 \) → \( x = 2 \)
- If \( y = 3 \): \( 4x = 1 \) → not valid

Valid pairs are:
- \( (x, y) = (7, 0) \)
- \( (x, y) = (2, 2) \)

6. **Counting the Combinations**:
For \( (x, y) = (7, 0) \):
- We have 7 entries as \( 2 \) and 2 entries as \( 0 \).
- The number of ways to choose positions for \( 2 \) and \( 0 \) is:
\[ \binom{9}{7} = \binom{9}{2} = 36 \]

For \( (x, y) = (2, 2) \):
- We have 2 entries as \( 2 \), 2 entries as \( 3 \), and 5 entries as \( 0 \).
- The number of ways to choose positions for \( 2 \), \( 3 \), and \( 0 \) is:
\[ \frac{9!}{2!2!5!} = \frac{362880}{2 \cdot 2 \cdot 120} = 126 \]

7. **Total Count**:
Adding the two cases gives:
\[ 36 + 126 = 162 \]

### Final Answer:
The total number of matrices \( X \) is **162**.