Answer: To determine whether the set of matrices S_k with fixed diagonal sum k is a subspace of the vector space of n x n matrices, we need to check three conditions:
- The set S_k is non-empty.
- If A and B are in S_k, then A + B is in S_k.
- If A is in S_k and c is a scalar, then cA is in S_k.
First, note that the zero matrix is always in S_k, since it has all diagonal entries equal to zero.
The set S_k is non-empty because it contains at least the zero matrix, which has diagonal sum 0.
Let A and B be two matrices in S_k. Then the diagonal entries of A + B are the sums of the corresponding diagonal entries of A and B. That is, the diagonal sum of A + B is:
diag(A + B) = diag(A) + diag(B) = k + k = 2k
Therefore, A + B is in S_{2k}, and hence in S_k. Thus, S_k is closed under addition.
Let A be a matrix in S_k and let c be a scalar. Then the diagonal entries of cA are c times the diagonal entries of A. That is, the diagonal sum of cA is:
diag(cA) = c diag(A) = c k
Therefore, cA is in S_{ck}, and hence in S_k. Thus, S_k is closed under scalar multiplication.
Since all three conditions are satisfied, we conclude that S_k is a subspace of the vector space of n x n matrices for any value of k.