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?}

• A. \( O \) is a point of local minimum of \( g \)
• B. \( O \) is a point of local maximum of \( g \)
• C. \( O \) is a saddle point of \( g \)
• D. \( O \) is not a critical point of \( g \)

Hint:

For quadratic functions, use the Hessian matrix to classify critical points as minima, maxima, or saddle points.

Answer 1

The Correct Option is C

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 \( {(3)} \).