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

Answer: (C)

The dimension of $S$ over $\mathbb{R}$ is 11.

Answer 2

The correct option is (C): The dimension of $S$ over $\mathbb{R}$ is 11.