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The image shows a matrix-related problem where you need to prove that \( A - A^T \) is a skew-symmetric matrix.

A skew-symmetric matrix \( B \) is defined by the property \( B^T = -B \). Therefore, to prove that \( A - A^T \) is skew-symmetric, we need to check whether:

\[
(A - A^T)^T = -(A - A^T)
\]

### Step-by-Step Explanation:

1. **Matrix \( A \) and its Transpose \( A^T \)**:
   You are given the matrix \( A \) and its transpose \( A^T \):

   \[
   A = \begin{pmatrix} 
   1 & 3 & -1 \\ 
   -3 & -5 & 8 \\ 
   1 & 8 & 9 
   \end{pmatrix}, 
   \quad
   A^T = \begin{pmatrix} 
   1 & -3 & 1 \\ 
   3 & -5 & 8 \\ 
   -1 & 8 & 9 
   \end{pmatrix}
   \]

2. **Subtraction \( A - A^T \)**:
   Let's compute the subtraction of these two matrices:

   \[
   A - A^T = \begin{pmatrix} 
   1 & 3 & -1 \\ 
   -3 & -5 & 8 \\ 
   1 & 8 & 9 
   \end{pmatrix}
   - 
   \begin{pmatrix} 
   1 & -3 & 1 \\ 
   3 & -5 & 8 \\ 
   -1 & 8 & 9 
   \end{pmatrix}
   \]

   Subtract corresponding elements:

   \[
   A - A^T = \begin{pmatrix} 
   1 - 1 & 3 - (-3) & -1 - 1 \\ 
   -3 - 3 & -5 - (-5) & 8 - 8 \\ 
   1 - (-1) & 8 - 8 & 9 - 9 
   \end{pmatrix}
   \]

   Simplifying the elements:

   \[
   A - A^T = \begin{pmatrix} 
   0 & 6 & -2 \\ 
   -6 & 0 & 0 \\ 
   2 & 0 & 0 
   \end{pmatrix}
   \]

3. **Check for Skew-Symmetry**:
   Now, check if this matrix is skew-symmetric by comparing it with its transpose:

   \[
   (A - A^T)^T = \begin{pmatrix} 
   0 & -6 & 2 \\ 
   6 & 0 & 0 \\ 
   -2 & 0 & 0 
   \end{pmatrix}
   \]

   Since \( (A - A^T)^T = -(A - A^T) \), the matrix is skew-symmetric, as required.

### Conclusion:
Thus, \( A - A^T \) is indeed a skew-symmetric matrix.

Would you like further details or have any questions?

### Related Questions:
1. What is the definition of a symmetric matrix?
2. How is the transpose of a matrix calculated?
3. Can a diagonal matrix be skew-symmetric?
4. What are the applications of skew-symmetric matrices in physics?
5. How does skew-symmetry affect eigenvalues of a matrix?

### Tip:
Always verify the skew-symmetric property by checking if the transpose of the matrix equals the negative of the original matrix.

Prove that A - A^T is a skew-symmetric matrix for the given matrix A.

Learn how to prove that A - A^T is a skew-symmetric matrix. This problem explores matrix transposition and subtraction, providing a detailed step-by-step solution. Suitable for students studying linear algebra at the university level.

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