Discussion
Consider the system of ordinary differential equations
\[ \frac{dX}{dt} = MX, \]
where M is a \(6 \times 6\) skew-symmetric matrix with entries in \(\mathbb{R}\). Then, for this system, the origin is a stable critical point for
Answer: A
The origin is stable for any such matrix M.
If M is skew-symmetric, then
\[ \frac{d}{dt}\|X\|^{2} = X^{T}(M + M^{T})X = 0 \]
so every solution keeps a constant length. A solution that starts within distance \(\varepsilon\) of the origin stays within \(\varepsilon\) for all time, which is Lyapunov stability. Equivalently, the eigenvalues of a real skew-symmetric matrix are purely imaginary (or zero) and the matrix is diagonalizable, so the solutions are bounded rotations.
The rank does not matter: a zero eigenvalue just gives constant components. The stability is not asymptotic, since solutions do not decay, but the question asks only for stability, so the rank-restricted options are wrong.
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