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  • Find All Values of $x$ so that a Matrix is SingularFind All Values of $x$ so that a Matrix is Singular Let \[A=\begin{bmatrix} 1 & -x & 0 & 0 \\ 0 &1 & -x & 0 \\ 0 & 0 & 1 & -x \\ 0 & 1 & 0 & -1 \end{bmatrix}\] be a $4\times 4$ matrix. Find all values of $x$ so that the matrix $A$ is singular.   Hint. Use the fact that a matrix is singular if and only […]
  • If a Group is of Odd Order, then Any Nonidentity Element is Not Conjugate to its InverseIf a Group is of Odd Order, then Any Nonidentity Element is Not Conjugate to its Inverse Let $G$ be a finite group of odd order. Assume that $x \in G$ is not the identity element. Show that $x$ is not conjugate to $x^{-1}$.   Proof. Assume the contrary, that is, assume that there exists $g \in G$ such that $gx^{-1}g^{-1}=x$. Then we have \[xg=gx^{-1}. […]
  • The Number of Elements Satisfying $g^5=e$ in a Finite Group is OddThe Number of Elements Satisfying $g^5=e$ in a Finite Group is Odd Let $G$ be a finite group. Let $S$ be the set of elements $g$ such that $g^5=e$, where $e$ is the identity element in the group $G$. Prove that the number of elements in $S$ is odd.   Proof. Let $g\neq e$ be an element in the group $G$ such that $g^5=e$. As […]
  • Construction of a Symmetric Matrix whose Inverse Matrix is ItselfConstruction of a Symmetric Matrix whose Inverse Matrix is Itself Let $\mathbf{v}$ be a nonzero vector in $\R^n$. Then the dot product $\mathbf{v}\cdot \mathbf{v}=\mathbf{v}^{\trans}\mathbf{v}\neq 0$. Set $a:=\frac{2}{\mathbf{v}^{\trans}\mathbf{v}}$ and define the $n\times n$ matrix $A$ by \[A=I-a\mathbf{v}\mathbf{v}^{\trans},\] where […]
  • Find a Quadratic Function Satisfying Conditions on DerivativesFind a Quadratic Function Satisfying Conditions on Derivatives Find a quadratic function $f(x) = ax^2 + bx + c$ such that $f(1) = 3$, $f'(1) = 3$, and $f^{\prime\prime}(1) = 2$. Here, $f'(x)$ and $f^{\prime\prime}(x)$ denote the first and second derivatives, respectively.   Solution. Each condition required on $f$ can be turned […]
  • Centralizer, Normalizer, and Center of the Dihedral Group $D_{8}$Centralizer, Normalizer, and Center of the Dihedral Group $D_{8}$ Let $D_8$ be the dihedral group of order $8$. Using the generators and relations, we have \[D_{8}=\langle r,s \mid r^4=s^2=1, sr=r^{-1}s\rangle.\] (a) Let $A$ be the subgroup of $D_8$ generated by $r$, that is, $A=\{1,r,r^2,r^3\}$. Prove that the centralizer […]
  • Prove the Cauchy-Schwarz InequalityProve the Cauchy-Schwarz Inequality Let $\mathbf{a}, \mathbf{b}$ be vectors in $\R^n$. Prove the Cauchy-Schwarz inequality: \[|\mathbf{a}\cdot \mathbf{b}|\leq \|\mathbf{a}\|\,\|\mathbf{b}\|.\]   We give two proofs. Proof 1 Let $x$ be a variable and consider the length of the vector […]
  • A Group of Order $20$ is SolvableA Group of Order $20$ is Solvable Prove that a group of order $20$ is solvable.   Hint. Show that a group of order $20$ has a unique normal $5$-Sylow subgroup by Sylow's theorem. See the post summary of Sylow’s Theorem to review Sylow's theorem. Proof. Let $G$ be a group of order $20$. The […]

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