## 每週問題 September 9, 2013

Let $\mathbb{R}^{n\times n}$ denote the vector space formed by all $n\times n$ real matrices. Determine which of the following subsets of $\mathbb{R}^{n\times n}$ are in fact subspaces of $\mathbb{R}^{n\times n}$.

(a) The diagonal matrices.
(b) The symmetric matrices.
(c) The nonsingular matrices.
(d) The singular matrices.
(e) The triangular matrices.
(f) The lower-triangular matrices.
(g) The idempotent matrices, i.e., $A^2=A$.
(h) The orthogonal matrices, i.e., $A^{-1}=A^T$.
(i) All matrices with zero trace.
(j) All matrices that commute with a given matrix $A$.