Question:
Let \( g : M_2({R}) \to {R \) be given by \( g(A) = \operatorname{Trace}(A^2) \). Let \( O \) be the \( 2 \times 2 \) zero matrix. The space \( M_2({R}) \) may be identified with \( {R}^4 \) in the usual manner. Which one of the following is correct?}
Let \( g : M_2({R}) \to {R \) be given by \( g(A) = \operatorname{Trace}(A^2) \). Let \( O \) be the \( 2 \times 2 \) zero matrix. The space \( M_2({R}) \) may be identified with \( {R}^4 \) in the usual manner. Which one of the following is correct?}
Show Hint
For quadratic functions, use the Hessian matrix to classify critical points as minima, maxima, or saddle points.
Updated On: Feb 1, 2025
- \( O \) is a point of local minimum of \( g \)
- \( O \) is a point of local maximum of \( g \)
- \( O \) is a saddle point of \( g \)
- \( O \) is not a critical point of \( g \)
Show Solution
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The Correct Option is C
Solution and Explanation
Step 1: Analyzing \( g(A) \).
The function \( g(A) = \operatorname{Trace}(A^2) \) depends on the eigenvalues of \( A \). At \( O \), \( g(A) = 0 \).
Step 2: Classifying the critical point.
Since \( g(A) \) can increase or decrease along different directions in \( M_2({R}) \), \( O \) is a saddle point.
Step 3: Conclusion.
The correct description of \( O \) is \( {(3)} \).
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