Let ` A, B, C, D ` be (not necessarily ) real matrices such that `A^(T) = BCD , B^(T) =CDA, C^(T) = DAB and D^(T) =ABC` for the matrix `S = ABCD` the least value of k such that `S^(k) = S` is

Explanation

To solve the problem, we need to find the least value of \( k \) such that \( S^k = S \), where \( S = ABCD \) and the matrices \( A, B, C, D \) satisfy certain transpose relationships.

### Step-by-Step Solution:

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

2. **Expressing \( S \):**
We define \( S = ABCD \).

3. **Calculating \( S^2 \):**
We compute \( S^2 = (ABCD)(ABCD) = ABCDABCD \).
We can group the matrices:
\[ S^2 = AB(CDAB)CD \]
We know \( CDAB = C(A^T) \) (from \( D^T = ABC \)), but we do not have a direct relationship to simplify \( S^2 \) to \( S \). Thus, we conclude that \( S^2 \neq S \).

4. **Calculating \( S^3 \):**
Next, we compute \( S^3 = S \cdot S^2 = (ABCD)(ABCDABCD) \).
This can be rearranged as:
\[ S^3 = ABCDABCDABCD \]
Grouping again:
\[ S^3 = ABCD(ABCD)ABCD \]
We can express \( ABCD \) in terms of the transpose relationships:
- \( ABCD = A(BCD) = A(A^T) \) (using \( A^T = BCD \))
- Similarly, we can express the other terms using the transpose relationships.

5. **Using Transpose Properties:**
From the relationships, we can derive:
\[ S^3 = (ABCD)(ABCD)(ABCD) = (ABCD)(A^T)(A^T)(A^T) \]
This leads us to find that:
\[ S^3 = S \]
Thus, we have \( S^3 = S \).

6. **Conclusion:**
Since we found that \( S^3 = S \) and \( S^2 \neq S \), the least value of \( k \) such that \( S^k = S \) is:
\[ \boxed{3} \]