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

The Correct Option is B

To determine the truth of statements \( P \) and \( Q \), we examine each one step-by-step.

Statement \( P \): \( (X, (Y, Z)) + (Y, (Z, X)) + (Z, (X, Y)) = 0 \) for all \( X, Y, Z \in M_2(\mathbb{R}) \).

This statement is related to the Jacobi identity, which commonly appears in the study of the Lie bracket. The given operation \((X, Y) = XY - YX\) is the matrix commutator, and the Jacobi identity for the commutator is:

\([(X, Y), Z] + [(Y, Z), X] + [(Z, X), Y] = 0\)

Computing these individually, we have:

• \((X, (Y, Z)) = X(YZ - ZY) - (YZ - ZY)X = XYZ - XZY - YZX + ZYX\)
• \((Y, (Z, X)) = Y(ZX - XZ) - (ZX - XZ)Y = YZX - YXZ - ZXY + XZY\)
• \((Z, (X, Y)) = Z(XY - YX) - (XY - YX)Z = ZXY - ZYX - XYZ + YXZ\)

Adding them together, we indeed find:

\((X, (Y, Z)) + (Y, (Z, X)) + (Z, (X, Y)) = (XYZ - XZY - YZX + ZYX) + (YZX - YXZ - ZXY + XZY) + (ZXY - ZYX - XYZ + YXZ) = 0\)

Thus, statement \( P \) is true.

Statement \( Q \): \( (X, (Y, Z)) = ((X, Y), Z) \) for all \( X, Y, Z \in M_2(\mathbb{R}) \).

Let's expand both sides to check for equality:

• Left Side: \((X, (Y, Z)) = X(YZ - ZY) - (YZ - ZY)X = XYZ - XZY - YZX + ZYX\)
• Right Side: XYZ - XZY - YZX + ZYX $\neq$ XYZ - YXZ - ZXY + ZYX

 

The results do not match, indicating that statement \( Q \) is false.

Conclusion: Based on our analysis, \( P \) is true, and \( Q \) is false. Therefore, the correct answer is that \( P \) is true but \( Q \) is false.