always zero
always pure imaginary
either zero or pure imaginary
always real
Answer is Right!
Answer is Wrong!
The correct answer is: C. either zero or pure imaginary.
A skew-symmetric matrix is a square matrix $A$ such that $A^T = -A$. The eigenvalues of a skew-symmetric matrix are always either zero or pure imaginary. This can be shown by using the fact that the characteristic polynomial of a skew-symmetric matrix is of the form $p(x) = x^2 – tr(A)x + \det(A)$. Since $tr(A) = 0$ for a skew-symmetric matrix, the characteristic polynomial has two distinct roots, which must be either both real or both complex. If the roots are real, then they must be both zero. If the roots are complex, then they must be a complex conjugate pair.
Here is a brief explanation of each option:
- Option A: The eigenvalues of a skew-symmetric matrix are always zero. This is not always true, as shown above.
- Option B: The eigenvalues of a skew-symmetric matrix are always pure imaginary. This is also not always true, as shown above.
- Option C: The eigenvalues of a skew-symmetric matrix are either zero or pure imaginary. This is the correct answer, as shown above.
- Option D: The eigenvalues of a skew-symmetric matrix are always real. This is not always true, as shown above.