Question

Let $P$ be the set of all non-singular matrices of order $3$ over $R$ and $Q$ be the set of all orthogonal matrices of order $3$ over $R$. Then

• A. $P$ is proper subset of $Q$
• B. $Q$ is proper subset of $P$
• C. Neither $P$ is proper subset of $Q$ nor $Q$ is proper subset of $P$
• D. $P\cap Q=\varphi$, the void set

Answer 1

The correct option is B $Q$ is proper subset of $P$

$P=\{A:|A|\in R-\{0\left\}\right\}$

$Q$ is orthogonal.
$\Rightarrow Q{Q}^T=I$
$\Rightarrow |Q{Q}^T|=|I|$
$\Rightarrow |Q{|}^2=1$
$\Rightarrow Q=\{A:|A|\in \{-1,1\left\}\right\}$

So, we can conclude $Q\subset P$