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• Simple Commutative Relation on Matrices Let $A$ and $B$ are $n \times n$ matrices with real entries. Assume that $A+B$ is invertible. Then show that \[A(A+B)^{-1}B=B(A+B)^{-1}A.\] (University of California, Berkeley Qualifying Exam) Proof. Let $P=A+B$. Then $B=P-A$. Using these, we express the given […]
• Coupon Collecting Problem: Find the Expectation of Boxes to Collect All Toys A box of some snacks includes one of five toys. The chances of getting any of the toys are equally likely and independent of the previous results. (a) Suppose that you buy the box until you complete all the five toys. Find the expected number of boxes that you need to buy. (b) […]
• A ring is Local if and only if the set of Non-Units is an Ideal A ring is called local if it has a unique maximal ideal. (a) Prove that a ring $R$ with $1$ is local if and only if the set of non-unit elements of $R$ is an ideal of $R$. (b) Let $R$ be a ring with $1$ and suppose that $M$ is a maximal ideal of $R$. Prove that if every […]
• 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 […]
• Using Properties of Inverse Matrices, Simplify the Expression Let $A, B, C$ be $n\times n$ invertible matrices. When you simplify the expression \[C^{-1}(AB^{-1})^{-1}(CA^{-1})^{-1}C^2,\] which matrix do you get? (a) $A$ (b) $C^{-1}A^{-1}BC^{-1}AC^2$ (c) $B$ (d) $C^2$ (e) $C^{-1}BC$ (f) $C$   Solution. In this problem, we […]
• Ring Homomorphisms and Radical Ideals Let $R$ and $R'$ be commutative rings and let $f:R\to R'$ be a ring homomorphism. Let $I$ and $I'$ be ideals of $R$ and $R'$, respectively. (a) Prove that $f(\sqrt{I}\,) \subset \sqrt{f(I)}$. (b) Prove that $\sqrt{f^{-1}(I')}=f^{-1}(\sqrt{I'})$ (c) Suppose that $f$ is […]
• Quiz 1. Gauss-Jordan Elimination / Homogeneous System. Math 2568 Spring 2017. (a) Solve the following system by transforming the augmented matrix to reduced echelon form (Gauss-Jordan elimination). Indicate the elementary row operations you performed. […]
• Sylow Subgroups of a Group of Order 33 is Normal Subgroups Prove that any $p$-Sylow subgroup of a group $G$ of order $33$ is a normal subgroup of $G$.   Hint. We use Sylow's theorem. Review the basic terminologies and Sylow's theorem. Recall that if there is only one $p$-Sylow subgroup $P$ of $G$ for a fixed prime $p$, then $P$ […]