GATE 2024 MA – Question 39

Calculus · Multivariable Differentiation and Optimization · 2 marks · Multiple choice

On M₂(ℝ)≅ℝ⁴, g(A)=trace(A²). Classify the zero matrix.

Local minimum
Local maximum
Saddle point
Not a critical point

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Correct answer: (C) Saddle point

Explanation

For $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$, g=a²+d²+2bc, with zero gradient at zero. Along tI, g=2t²>0; along $\begin{pmatrix}0&t\\-t&0\end{pmatrix}$, g=−2t²<0. Both signs occur arbitrarily near zero, so **saddle, C**.

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