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?
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?
\[ P : (X, (Y, Z)) + (Y, (Z, X)) + (Z, (X, Y)) = 0 \text{ for all } X, Y, Z \in M_2(\mathbb{R}).\]\[ Q : (X, (Y, Z)) = ((X, Y), Z) \text{ for all } X, Y, Z \in M_2(\mathbb{R}).\]Then which one of the following is correct?
Updated On: Oct 1, 2024
- Both \( P \) and \( Q \) are true.
- \( P \) is true, but \( Q \) is false.
- \( P \) is false, but \( Q \) is true.
- Both \( P \) and \( Q \) are false.
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The Correct Option is B
Solution and Explanation
The correct option is (B): \( P \) is true, but \( Q \) is false.
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