Let A,B,C,D be real matrices such that `A^(T)=BCD,B^(T)=CDA,C^(T)=DAB and D^(T)=ABC` for the matrix M=ABCD then find `M^(3)`?

Options

A. M
B. `M^(2)`
C. `M^(3)`
D. `M^(4)`

Answer

To solve the problem, we need to find \( M^3 \) for the matrix \( M = ABCD \) given the relationships between the matrices \( A, B, C, \) and \( D \).

### Step-by-Step Solution:

1. **Understanding the Transpose Relationships**:
We are given the following relationships:
- \( A^T = BCD \)
- \( B^T = CDA \)
- \( C^T = DAB \)
- \( D^T = ABC \)

2. **Finding the Transpose of \( M \)**:
We start with the matrix \( M \):
\[ M = ABCD \]
To find \( M^T \), we use the property of transposes:
\[ M^T = (ABCD)^T = D^T C^T B^T A^T \]

3. **Substituting the Transpose Relationships**:
Now we substitute the transpose relationships into the equation:
- Substitute \( D^T = ABC \)
- Substitute \( C^T = DAB \)
- Substitute \( B^T = CDA \)
- Substitute \( A^T = BCD \)

Thus, we have:
\[ M^T = (ABC)(DAB)(CDA)(BCD) \]

4. **Rearranging the Terms**:
Now we can rearrange the terms:
\[ M^T = ABCD \cdot ABCD \cdot ABCD = M \cdot M \cdot M = M^3 \]

5. **Conclusion**:
From the above steps, we find that:
\[ M^T = M^3 \]
This means that \( M^3 \) is equal to the transpose of \( M \).

### Final Result:
Thus, we conclude that:
\[ M^3 = M^T \]