Question 103
Let $A \in \mathcal{M}_n(\mathbb{C})$ be a matrix with the property $A^T = -A$, where $A^T$ is the transpose of $A$.
a) If $A \in \mathcal{M}_n(\mathbb{R})$ and $A^2 = O_n$, prove that $A = O_n$.
b) If $n$ is an odd natural number and there is a matrix $B \in \mathcal{M}_n(\mathbb{C})$ such that $A$ is the adjoint of $B$, prove that $A^2 = O_n$.
a) If $A \in \mathcal{M}_n(\mathbb{R})$ and $A^2 = O_n$, prove that $A = O_n$.
b) If $n$ is an odd natural number and there is a matrix $B \in \mathcal{M}_n(\mathbb{C})$ such that $A$ is the adjoint of $B$, prove that $A^2 = O_n$.