Problem in skew-symmetric matrix 
Let $A$ be a real skew-symmetric matrix. Prove that $I+A$ is non-singular, where $I$ is the identity matrix. 


Here is a proof using the geometric meaning of a skew-symmetric matrix in 3D space.
For an arbitrary vector $a=[a_1,a_2,a_3]^T\in\mathbb{R}^3$, there exists a skew-symmetric matrix
$$[a]_{\times}=\left( \begin{array}{ccc} 
0 & -a_3 & a_2 \\
a_3 & 0 & -a_1 \\
-a_2 & a_1 & 0 \\
\end{array} \right).$$
Conversely, given an arbitrary skew-symmetric matrix $A$, there is an associating vector $a$ such that $A=[a]_{\times}$.
Given a skew-symmetric matrix $A$ and a vector $x$, we have $Ax=a\times x$.
Since $a\times x \perp kx, \forall k\ne 0$, it is impossible $Ax=kx, \forall k\ne 0$. Therefore, $kI+A, \forall k\ne 0$ is nonsingular. 
