Let $G$ be a finite group of order $p^n$, where $p$ is a prime number and $n$ is a positive integer.
Suppose that $H$ is a subgroup of $G$ with index $[G:P]=p$.
Then prove that $H$ is a normal subgroup of $G$.

(Michigan State University, Abstract Algebra Qualifying Exam)

Let $A$ be a real symmetric $n\times n$ matrix with $0$ as a simple eigenvalue (that is, the algebraic multiplicity of the eigenvalue $0$ is $1$), and let us fix a vector $\mathbf{v}\in \R^n$.

(a) Prove that for sufficiently small positive real $\epsilon$, the equation
\[A\mathbf{x}+\epsilon\mathbf{x}=\mathbf{v}\]
has a unique solution $\mathbf{x}=\mathbf{x}(\epsilon) \in \R^n$.

(b) Evaluate
\[\lim_{\epsilon \to 0^+} \epsilon \mathbf{x}(\epsilon)\]
in terms of $\mathbf{v}$, the eigenvectors of $A$, and the inner product $\langle\, ,\,\rangle$ on $\R^n$.

(University of California, Berkeley, Linear Algebra Qualifying Exam)

(a) Is it true that $A$ must commute with its transpose?

(b) Suppose that the columns of $A$ (considered as vectors) form an orthonormal set.
Is it true that the rows of $A$ must also form an orthonormal set?

(University of California, Berkeley, Linear Algebra Qualifying Exam)

Prove that the matrix
\[A=\begin{bmatrix}
1 & 1.00001 & 1 \\
1.00001 &1 &1.00001 \\
1 & 1.00001 & 1
\end{bmatrix}\]
has one positive eigenvalue and one negative eigenvalue.

(University of California, Berkeley Qualifying Exam Problem)

Let $A$ be an $n \times n$ matrix over a field $K$. Prove that
\[\rk(A^2)-\rk(A^3)\leq \rk(A)-\rk(A^2),\]
where $\rk(B)$ denotes the rank of a matrix $B$.

(University of California, Berkeley, Qualifying Exam)