For a real skew-symmetric matrix $A$ ($A^T = -A$), which of the following statements must be TRUE?
Options:
(A) All diagonal elements of $A$ are zero
(B) All eigenvalues of $A$ are purely imaginary or zero
(C) The determinant of $A$ is zero if $A$ is of odd order
(D) The matrix $A$ is always invertible
Answer Key:
A, B, C
Step-by-Step Solution & Explanation
- A is true: $a_{ii} = -a_{ii} \implies a_{ii} = 0$.
- B is true: For skew-symmetric real matrices, eigenvalues are of the form $i \beta$ (purely imaginary or 0).
- C is true: $\det(A) = \det(A^T) = \det(-A) = (-1)^n \det(A)$. For odd $n$, $\det(A) = -\det(A) \implies \det(A) = 0$.
- D is false: Any odd-order skew-symmetric matrix has determinant zero and is singular (non-invertible).