Let A and B be two square matrices of the same size such that `AB^(T)+BA^(T)=O`. If A is a skew-symmetric matrix then BA is

Options

A. a symmetric matrix
B. a skew-symmetric matrix
C. an orthogonal matrix
D. an invertible matrix

Explanation

To solve the problem, we need to analyze the given information step by step.

### Step-by-Step Solution:

**Step 1: Understand the properties of skew-symmetric matrices.**
A matrix \( A \) is skew-symmetric if \( A^T = -A \). This property will be useful in our calculations.

**Step 2: Use the given equation.**
We are given that:
\[ AB^T + BA^T = 0 \]
This can be rearranged to:
\[ AB^T = -BA^T \]

**Step 3: Substitute the property of skew-symmetric matrices.**
Since \( A \) is skew-symmetric, we have:
\[ A^T = -A \]
Substituting this into our equation gives:
\[ AB^T = -B(-A) = BA \]
Thus, we have:
\[ AB^T = BA \]

**Step 4: Analyze the expression for \( BA \).**
From the equation \( AB^T = BA \), we can denote \( E = BA \). Therefore:
\[ E = AB^T \]

**Step 5: Take the transpose of both sides.**
Now, we take the transpose of \( E \):
\[ E^T = (BA)^T = A^T B^T \]
Using the property of skew-symmetry again, we substitute \( A^T \):
\[ E^T = (-A)B^T = -AB^T \]

**Step 6: Substitute back into the equation.**
Since we know that \( AB^T = E \), we can substitute:
\[ E^T = -E \]

**Step 7: Conclude the nature of \( E \).**
The equation \( E^T = -E \) indicates that \( E \) is skew-symmetric. Therefore, we conclude:
\[ BA \text{ is a skew-symmetric matrix.} \]

### Final Answer:
Thus, if \( A \) is a skew-symmetric matrix, then \( BA \) is also a skew-symmetric matrix.