# Harvard-University-exam-eye-catch

### More from my site

• Find 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 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 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 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 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 $D_{8}$ 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 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 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 […]