Question

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

• A. any such matrix \( M \)
• B. only such matrices \( M \) whose rank is 2
• C. only such matrices \( M \) whose rank is 4
• D. only such matrices \( M \) whose rank is 6

Hint:

For systems with skew-symmetric matrices, the origin is always a stable critical point because the eigenvalues of a skew-symmetric matrix are purely imaginary.

Answer 1

The Correct Option is A

In this system, \( M \) is a skew-symmetric matrix. A skew-symmetric matrix has purely imaginary eigenvalues. The stability of the critical point at the origin depends on the eigenvalues of the matrix \( M \).

Since the eigenvalues of a skew-symmetric matrix are purely imaginary, the origin will always be a center and will be a stable critical point. The stability condition holds regardless of the rank of \( M \).

Thus, the origin is a stable critical point for any skew-symmetric matrix \( M \).

\[ \boxed{A} \quad \text{any such matrix } M. \]