7. Let $B = \left( \left( b _ { i , j } \right) \right)$ be an $n \times n$ matrix. Let $p : \{ 1,2 , \ldots , n \} \mapsto \{ 1,2 , \ldots , n \}$ be a bijection (i.e. a one-to-one correspondence) and let a matrix $A = \left( \left( a _ { i , j } \right) \right)$ be defined by $$a _ { i , j } = b _ { p ( i ) , p ( j ) } , \quad 1 \leq i , j \leq n .$$ Which of the following statement(s) is/are true for all choices of $B$ and $P$. (a) $A$ admits an inverse if and only if $B$ admits an inverse. (b) For any $x , y \in \mathbb { R } ^ { n } , A x = y$ admits a solution if and only if $B x = y$ admits a solution. (c) $A$ and $B$ have the same trace. (d) $A$ and $B$ have the same eigenvectors.
7. Let $B = \left( \left( b _ { i , j } \right) \right)$ be an $n \times n$ matrix. Let $p : \{ 1,2 , \ldots , n \} \mapsto \{ 1,2 , \ldots , n \}$ be a bijection (i.e. a one-to-one correspondence) and let a matrix $A = \left( \left( a _ { i , j } \right) \right)$ be defined by
$$a _ { i , j } = b _ { p ( i ) , p ( j ) } , \quad 1 \leq i , j \leq n .$$
Which of the following statement(s) is/are true for all choices of $B$ and $P$.\\
(a) $A$ admits an inverse if and only if $B$ admits an inverse.\\
(b) For any $x , y \in \mathbb { R } ^ { n } , A x = y$ admits a solution if and only if $B x = y$ admits a solution.\\
(c) $A$ and $B$ have the same trace.\\
(d) $A$ and $B$ have the same eigenvectors.\\