Real and symmetric matrices

Question

Two questions regarding real symmetric matrices.

1) $A$, $B$, $C$, $D$ are $n\times n$ real matrices, $A \sim B$ and $C \sim D$. Then $AC\sim BD$ ($\sim$ = similar).

• I know if $A \sim B$ then $A = PBP^{-1}$. But I believe $AC\sim BD$ is not true.

2) If $A$ and $B$ are $n\times n$ real symmetric matrices and congruent then they are also similar.

• I know if they are congruent then there exists an invertible matrix $P$ such that $P^TBP = A$ and they have the same rank. I see that $P^T = P^{-1}$ then it is orthogonal also. But not sure if "similar".

Answer 1

Hints.

1. Consider $A=B=C=\pmatrix{0&1\\ 0&0}$. Find a matrix $D$ to disprove the statement.

2. Similar matrices must have identical determinants. Now, consider the $1\times1$ case. The determinant of a $1\times1$ matrix (i.e. a scalar) is simply the value of the scalar. Is it possible to find two real scalars $A, B$ such that $A=P^TBP$ for some $P\ne0$ but $A\ne B$?