# top10mathproblems2017

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• Subgroup of Finite Index Contains a Normal Subgroup of Finite Index Let $G$ be a group and let $H$ be a subgroup of finite index. Then show that there exists a normal subgroup $N$ of $G$ such that $N$ is of finite index in $G$ and $N\subset H$.   Proof. The group $G$ acts on the set of left cosets $G/H$ by left multiplication. Hence […]
• Rotation Matrix in Space and its Determinant and Eigenvalues For a real number $0\leq \theta \leq \pi$, we define the real $3\times 3$ matrix $A$ by $A=\begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta &\cos\theta &0 \\ 0 & 0 & 1 \end{bmatrix}.$ (a) Find the determinant of the matrix $A$. (b) Show that $A$ is an […]
• Find All 3 by 3 Reduced Row Echelon Form Matrices of Rank 1 and 2 (a) Find all $3 \times 3$ matrices which are in reduced row echelon form and have rank 1. (b) Find all such matrices with rank 2.   Solution. (a) Find all $3 \times 3$ matrices which are in reduced row echelon form and have rank 1. First we look at the rank 1 case. […]
• $\sqrt[m]{2}$ is an Irrational Number Prove that $\sqrt[m]{2}$ is an irrational number for any integer $m \geq 2$.   Hint. Use ring theory: Consider the polynomial $f(x)=x^m-2$. Apply Eisenstein's criterion, show that $f(x)$ is irreducible over $\Q$. Proof. Consider the monic polynomial […]
• Normal Nilpotent Matrix is Zero Matrix A complex square ($n\times n$) matrix $A$ is called normal if $A^* A=A A^*,$ where $A^*$ denotes the conjugate transpose of $A$, that is $A^*=\bar{A}^{\trans}$. A matrix $A$ is said to be nilpotent if there exists a positive integer $k$ such that $A^k$ is the zero […]
• Diagonalize a 2 by 2 Matrix if Diagonalizable Determine whether the matrix $A=\begin{bmatrix} 1 & 4\\ 2 & 3 \end{bmatrix}$ is diagonalizable. If so, find a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$. (The Ohio State University, Linear Algebra Final Exam […]
• If Squares of Elements in a Group Lie in a Subgroup, then It is a Normal Subgroup Let $H$ be a subgroup of a group $G$. Suppose that for each element $x\in G$, we have $x^2\in H$. Then prove that $H$ is a normal subgroup of $G$. (Purdue University, Abstract Algebra Qualifying Exam)   Proof. To show that $H$ is a normal subgroup of […]
• Find All Symmetric Matrices satisfying the Equation Find all $2\times 2$ symmetric matrices $A$ satisfying $A\begin{bmatrix} 1 \\ -1 \end{bmatrix} = \begin{bmatrix} 2 \\ 3 \end{bmatrix}$? Express your solution using free variable(s).   Solution. Let $A=\begin{bmatrix} a & b\\ c& d \end{bmatrix}$ […]