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. 


Let $\lambda \neq 0$ be an eigenvalue of $A$ with eigenvector $x$. Then:
$$x^* A x = x^*(\lambda x) = \lambda x^* x$$
Where $x^*$ is the hermitian adjoint. Now, since $A$ is real, $A^* = A^T$ and we get $(x^* A x)^* = x^* A^T x = -x^* A x$. And since $(x^* A x)^* = (\lambda x^* x)^* = \lambda^* x^* x$. Putting these two equations together yields:
$$x^* A x = - \lambda^* x^* x$$
But since we have the same vector $x$, we have $-\lambda^* = \lambda$. Now, say $\lambda = a + ib$, so $-\lambda^* = -a + ib$. Thus we get $\lambda = ib$, i.e. $\lambda$ is pure imaginary.
Now, say we have an eigenvalue $\lambda = ib$ with eigenvector $x$, $Ax = \lambda x$. This implies $(Ax^*) = \lambda^* x^*$ and $(Ax^*) = x^* A^* = x^* A^T = - x^* A$, so $-x^* A = \lambda^* x^*$ Take the tranpose of both sides $(-x^* A)^T = -A^T \overline{x} = A \overline{x}$ and $(\lambda^* x^*)^T = \lambda^* \overline{x}$, where $\overline{x}$ is the complex conjugate of $x$. We have reached:
$$A \overline{x} = \lambda^* \overline{x}$$
Thus, $\lambda^* = -ib$ is an eigenvalue to $A$ with eigenvector $\overline{x}$.
So, all non-zero eigenvalues of a real skew symmetric matrix are pure imaginary and come in pairs $\lambda$ and $-\lambda$. 
Now, let $\lambda$ be a (possibly zero) eigenvalue of $A$ with eigenvector $x$. From this eigenvalue we get an eigenvalue for $I + A$:
$$(I+A)x = Ix + Ax = x + \lambda x = (1+\lambda)x$$
since $\lambda$ is pure imaginary or zero, $1 + \lambda$ will always be non-zero. Since the determinant of a matrix is the product of its eigenvalues, we have that $\det(I+A) \neq 0$ and we can even deduce that $\det (I+A)$ is real and positive (since $(1 + ib)(1-ib) = 1 + b^2$). Hence $I+A$ is always invertible. 
Just a note: If $A$ is $n \times n$ with $n$ odd, $A$ will always have a zero eigenvalue, since
$$\det A = \det A^T = \det (-A) = (-1)^n \det A$$
if $n$ is odd we have $\det A = - \det A$ which implies $\det A = 0$, which implies that at least one eigenvalue is zero.
