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?
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?
Updated On: Oct 1, 2024
- \( S \) is a subspace of the vector space over \(\mathbb{R}\) of all \( 5 \times 5 \) symmetric matrices.
- The dimension of \( S \) over \(\mathbb{R}\) is 5.
- The dimension of \( S \) over \(\mathbb{R}\) is 11.
- If \( P \in S \) and all the entries of \( P \) are integers, then 5 divides the sum of all the diagonal entries of \( P \).
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The Correct Option is C
Solution and Explanation
The correct option is (C): The dimension of \( S \) over \(\mathbb{R}\) is 11.
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