For a $2 \times 2$ non-zero matrix $P$ with real entries, let $s_k$ be the sum of the diagonal entries of $P^k$ for any positive integer $k$. Then

• Correct answer
  A

  $P$ is a singular matrix if $s_k=0$ for every $k \geq 1$
• Correct answer
  B

  $s_{2 m} \gt 0$ for every positive integer $m \geq 2$ if $P$ is non-singular and $s_1=0$
• Correct answer
  C

  $s_{2019}=0$ if $s_1=0$
• D

  there exists a $2 \times 2$ matrix $Q$ such that $P Q-Q P=I$, where $I$ is the $2 \times 2$ identity matrix

Hence, the correct answer is options: 'A', 'B', 'C'.