• Find Values of $a, b, c$ such that the Given Matrix is Diagonalizable For which values of constants $a, b$ and $c$ is the matrix $A=\begin{bmatrix} 7 & a & b \\ 0 &2 &c \\ 0 & 0 & 3 \end{bmatrix}$ diagonalizable? (The Ohio State University, Linear Algebra Final Exam Problem)   Solution. Note that the […]
• Perturbation of a Singular Matrix is Nonsingular Suppose that $A$ is an $n\times n$ singular matrix. Prove that for sufficiently small $\epsilon>0$, the matrix $A-\epsilon I$ is nonsingular, where $I$ is the $n \times n$ identity matrix. Hint. Consider the characteristic polynomial $p(t)$ of the matrix $A$. Note […]
• Calculate Determinants of Matrices Calculate the determinants of the following $n\times n$ matrices. $A=\begin{bmatrix} 1 & 0 & 0 & \dots & 0 & 0 &1 \\ 1 & 1 & 0 & \dots & 0 & 0 & 0 \\ 0 & 1 & 1 & \dots & 0 & 0 & 0 \\ \vdots & \vdots […] • Polynomial (x-1)(x-2)\cdots (x-n)-1 is Irreducible Over the Ring of Integers \Z For each positive integer n, prove that the polynomial \[(x-1)(x-2)\cdots (x-n)-1$ is irreducible over the ring of integers $\Z$.   Proof. Note that the given polynomial has degree $n$. Suppose that the polynomial is reducible over $\Z$ and it decomposes as […]
• The Product of a Subgroup and a Normal Subgroup is a Subgroup Let $G$ be a group. Let $H$ be a subgroup of $G$ and let $N$ be a normal subgroup of $G$. The product of $H$ and $N$ is defined to be the subset $H\cdot N=\{hn\in G\mid h \in H, n\in N\}.$ Prove that the product $H\cdot N$ is a subgroup of […]
• Eckmann–Hilton Argument: Group Operation is a Group Homomorphism Let $G$ be a group with the identity element $e$ and suppose that we have a group homomorphism $\phi$ from the direct product $G \times G$ to $G$ satisfying $\phi(e, g)=g \text{ and } \phi(g, e)=g, \tag{*}$ for any $g\in G$. Let $\mu: G\times G \to G$ be a map defined […]
• If the Order of a Group is Even, then the Number of Elements of Order 2 is Odd Prove that if $G$ is a finite group of even order, then the number of elements of $G$ of order $2$ is odd.   Proof. First observe that for $g\in G$, $g^2=e \iff g=g^{-1},$ where $e$ is the identity element of $G$. Thus, the identity element $e$ and the […]
• 7 Problems on Skew-Symmetric Matrices Let $A$ and $B$ be $n\times n$ skew-symmetric matrices. Namely $A^{\trans}=-A$ and $B^{\trans}=-B$. (a) Prove that $A+B$ is skew-symmetric. (b) Prove that $cA$ is skew-symmetric for any scalar $c$. (c) Let $P$ be an $m\times n$ matrix. Prove that $P^{\trans}AP$ is […]