Let A be a `3xx3` matrix given by ` A = [a_(ij)].` If for every
column vector `X, X^(T)AX=O and a_(23)=-1008, ` the sum
of the digits of `a_(32)` is

Explanation

To solve the problem, we start by analyzing the given information about the matrix \( A \) and the condition \( X^T A X = 0 \) for every column vector \( X \).

### Step-by-Step Solution:

1. **Understanding the Condition**:
The condition \( X^T A X = 0 \) for every vector \( X \) implies that the matrix \( A \) is a skew-symmetric matrix. This means that \( A^T = -A \).

2. **Matrix Representation**:
Let \( A \) be represented as:
\[ A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} \]

3. **Using the Skew-Symmetric Property**:
From the skew-symmetric property, we have:
\[ a_{ij} = -a_{ji} \]
Therefore, we can write the relationships:
- \( a_{21} = -a_{12} \)
- \( a_{31} = -a_{13} \)
- \( a_{32} = -a_{23} \)
- \( a_{33} = 0 \) (since \( a_{33} = -a_{33} \))

4. **Given Information**:
We are given that \( a_{23} = -1008 \).

5. **Finding \( a_{32} \)**:
Using the relationship derived from the skew-symmetric property:
\[ a_{32} = -a_{23} \]
Substituting the value of \( a_{23} \):
\[ a_{32} = -(-1008) = 1008 \]

6. **Calculating the Sum of the Digits of \( a_{32} \)**:
Now, we need to find the sum of the digits of \( a_{32} = 1008 \).
- The digits of \( 1008 \) are \( 1, 0, 0, 8 \).
- The sum of these digits is:
\[ 1 + 0 + 0 + 8 = 9 \]

### Final Answer:
The sum of the digits of \( a_{32} \) is \( 9 \).

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