Question

For \( P \in M_5(\mathbb{R}) \) and \( i,j \in \{1,2, \ldots, 5\} \), let \( p_{ij} \) denote the \((i,j)\)th entry of \( P \). Let\[ S = \{ P \in M_5(\mathbb{R}) : p_{ij} = p_{sr} \text{ for } i,j,s,r \in \{1,2, \ldots, 5\} \text{ with } i + r = j + s \}.\]Then which one of the following is FALSE?

• A. \( S \) is a subspace of the vector space over \(\mathbb{R}\) of all \( 5 \times 5 \) symmetric matrices.
• B. The dimension of \( S \) over \(\mathbb{R}\) is 5.
• C. The dimension of \( S \) over \(\mathbb{R}\) is 11.
• D. If \( P \in S \) and all the entries of \( P \) are integers, then 5 divides the sum of all the diagonal entries of \( P \).

Answer 1

The Correct Option is C

To solve this problem, we need to understand the properties of the set \( S \) defined in the problem and analyze the correctness of each statement, particularly focusing on the dimensions of this set.

The set \( S = \{ P \in M_5(\mathbb{R}) : p_{ij} = p_{sr} \text{ for } i,j,s,r \in \{1,2, \ldots, 5\} \text{ with } i + r = j + s \} \)

This set \( S \) comprises matrices where entries with equal sums of their row and column indices are identical. In other words, all entries \( p_{ij} \) along diagonals where \( i + j \) is constant must be the same.

1. Understanding the structure of matrices in \( S \):

   For a \( 5 \times 5 \) matrix, the constant sums for indices range from 2 (when \( i = 1, j = 1 \)) to 10 (when \( i = 5, j = 5 \)). Therefore, there are 9 distinct "constant sum" diagonals, i.e., five \( p_{11}, p_{12}, p_{21} \), etc., up to \( p_{55} \).

2. Determining the dimension of \( S \):

   Each "constant sum" diagonal can have a unique entry. Thus, each matrix in \( S \) is determined by 9 independent parameters (one entry for each "constant sum"). Consequently, the dimension of \( S \) over \(\mathbb{R}\) is 9.

3. Analyzing the given statements:
   • S is a subspace of the vector space of all \( 5 \times 5 \) symmetric matrices: Each matrix in \( S \) can indeed be symmetric based on the described condition.
   • The dimension of \( S \) over \(\mathbb{R}\) is 5: Incorrect, as discussed, it is 9.
   • The dimension of \( S \) over \(\mathbb{R}\) is 11: Incorrect, as stated before, it should be 9.
   • If \( P \in S \) and all the entries of \( P \) are integers, then 5 divides the sum of all the diagonal entries of \( P \): This statement is true because along each "constant sum" diagonal, entries will be such that \( i+j \) remains constant, and there are a total of 25 entries.

Therefore, the statement "The dimension of \( S \) over \(\mathbb{R}\) is 11." is falsely stated in the context of dimensions because the dimension is actually 9.