Question

For $X, Y \in M_2(\mathbb{R})$, define $(X, Y) = XY - YX$. Let $0 \in M_2(\mathbb{R})$ denote the zero matrix. Consider the two statements:
P : (X, (Y, Z)) + (Y, (Z, X)) + (Z, (X, Y)) = 0 \text{ for all } X, Y, Z \in M_2(\mathbb{R}).$$$$ Q : (X, (Y, Z)) = ((X, Y), Z) \text{ for all } X, Y, Z \in M_2(\mathbb{R}).Then which one of the following is correct?

• A. Both $P$ and $Q$ are true.
• B. $P$ is true, but $Q$ is false.
• C. $P$ is false, but $Q$ is true.
• D. Both $P$ and $Q$ are false.

Answer 1

Answer: (B)

$P$ is true, but $Q$ is false.

Answer 2

The correct option is (B): $P$ is true, but $Q$ is false.