# Harvard-University-exam-eye-catch

by Yu ·

Add to solve later

Sponsored Links

Add to solve later

Sponsored Links

Add to solve later

Sponsored Links

### More from my site

- Use the Cayley-Hamilton Theorem to Compute the Power $A^{100}$ Let $A$ be a $3\times 3$ real orthogonal matrix with $\det(A)=1$. (a) If $\frac{-1+\sqrt{3}i}{2}$ is one of the eigenvalues of $A$, then find the all the eigenvalues of $A$. (b) Let \[A^{100}=aA^2+bA+cI,\] where $I$ is the $3\times 3$ identity matrix. Using the […]
- The Center of a p-Group is Not Trivial Let $G$ be a group of order $|G|=p^n$ for some $n \in \N$. (Such a group is called a $p$-group.) Show that the center $Z(G)$ of the group $G$ is not trivial. Hint. Use the class equation. Proof. If $G=Z(G)$, then the statement is true. So suppose that $G\neq […]
- The Symmetric Group is a Semi-Direct Product of the Alternating Group and a Subgroup $\langle(1,2) \rangle$ Prove that the symmetric group $S_n$, $n\geq 3$ is a semi-direct product of the alternating group $A_n$ and the subgroup $\langle(1,2) \rangle$ generated by the element $(1,2)$. Definition (Semi-Direct Product). Internal Semi-Direct-Product Recall that a group $G$ is […]
- Polynomial Ring with Integer Coefficients and the Prime Ideal $I=\{f(x) \in \Z[x] \mid f(-2)=0\}$ Let $\Z[x]$ be the ring of polynomials with integer coefficients. Prove that \[I=\{f(x)\in \Z[x] \mid f(-2)=0\}\] is a prime ideal of $\Z[x]$. Is $I$ a maximal ideal of $\Z[x]$? Proof. Define a map $\phi: \Z[x] \to \Z$ defined by \[\phi \left( f(x) […]
- Group of Invertible Matrices Over a Finite Field and its Stabilizer Let $\F_p$ be the finite field of $p$ elements, where $p$ is a prime number. Let $G_n=\GL_n(\F_p)$ be the group of $n\times n$ invertible matrices with entries in the field $\F_p$. As usual in linear algebra, we may regard the elements of $G_n$ as linear transformations on $\F_p^n$, […]
- Determinant of a General Circulant Matrix Let \[A=\begin{bmatrix} a_0 & a_1 & \dots & a_{n-2} &a_{n-1} \\ a_{n-1} & a_0 & \dots & a_{n-3} & a_{n-2} \\ a_{n-2} & a_{n-1} & \dots & a_{n-4} & a_{n-3} \\ \vdots & \vdots & \dots & \vdots & \vdots \\ a_{2} & a_3 & \dots & a_{0} & a_{1}\\ a_{1} & a_2 & […]
- Common Eigenvector of Two Matrices and Determinant of Commutator Let $A$ and $B$ be $n\times n$ matrices. Suppose that these matrices have a common eigenvector $\mathbf{x}$. Show that $\det(AB-BA)=0$. Steps. Write down eigenequations of $A$ and $B$ with the eigenvector $\mathbf{x}$. Show that AB-BA is singular. A matrix is […]
- Group Generated by Commutators of Two Normal Subgroups is a Normal Subgroup Let $G$ be a group and $H$ and $K$ be subgroups of $G$. For $h \in H$, and $k \in K$, we define the commutator $[h, k]:=hkh^{-1}k^{-1}$. Let $[H,K]$ be a subgroup of $G$ generated by all such commutators. Show that if $H$ and $K$ are normal subgroups of $G$, then the subgroup […]