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Math-Magic


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  • Every Integral Domain Artinian Ring is a FieldEvery Integral Domain Artinian Ring is a Field Let $R$ be a ring with $1$. Suppose that $R$ is an integral domain and an Artinian ring. Prove that $R$ is a field.   Definition (Artinian ring). A ring $R$ is called Artinian if it satisfies the defending chain condition on ideals. That is, whenever we have […]
  • Determine a 2-Digit Number Satisfying Two ConditionsDetermine a 2-Digit Number Satisfying Two Conditions A 2-digit number has two properties: The digits sum to 11, and if the number is written with digits reversed, and subtracted from the original number, the result is 45. Find the number.   Solution. The key to this problem is noticing that our 2-digit number can be […]
  • If Two Subsets $A, B$ of a Finite Group $G$ are Large Enough, then $G=AB$If Two Subsets $A, B$ of a Finite Group $G$ are Large Enough, then $G=AB$ Let $G$ be a finite group and let $A, B$ be subsets of $G$ satisfying \[|A|+|B| > |G|.\] Here $|X|$ denotes the cardinality (the number of elements) of the set $X$. Then prove that $G=AB$, where \[AB=\{ab \mid a\in A, b\in B\}.\]   Proof. Since $A, B$ […]
  • Subspaces of the Vector Space of All Real Valued Function on the IntervalSubspaces of the Vector Space of All Real Valued Function on the Interval Let $V$ be the vector space over $\R$ of all real valued functions defined on the interval $[0,1]$. Determine whether the following subsets of $V$ are subspaces or not. (a) $S=\{f(x) \in V \mid f(0)=f(1)\}$. (b) $T=\{f(x) \in V \mid […]
  • If a Matrix is the Product of Two Matrices, is it Invertible?If a Matrix is the Product of Two Matrices, is it Invertible? (a) Let $A$ be a $6\times 6$ matrix and suppose that $A$ can be written as \[A=BC,\] where $B$ is a $6\times 5$ matrix and $C$ is a $5\times 6$ matrix. Prove that the matrix $A$ cannot be invertible. (b) Let $A$ be a $2\times 2$ matrix and suppose that $A$ can be […]
  • For Which Choices of $x$ is the Given Matrix Invertible?For Which Choices of $x$ is the Given Matrix Invertible? Determine the values of $x$ so that the matrix \[A=\begin{bmatrix} 1 & 1 & x \\ 1 &x &x \\ x & x & x \end{bmatrix}\] is invertible. For those values of $x$, find the inverse matrix $A^{-1}$.   Solution. We use the fact that a matrix is invertible […]
  • Does the Trace Commute with Matrix Multiplication? Is $\tr (A B) = \tr (A) \tr (B) $?Does the Trace Commute with Matrix Multiplication? Is $\tr (A B) = \tr (A) \tr (B) $? Let $A$ and $B$ be $n \times n$ matrices. Is it always true that $\tr (A B) = \tr (A) \tr (B) $? If it is true, prove it. If not, give a counterexample.   Solution. There are many counterexamples. For one, take \[A = \begin{bmatrix} 1 & 0 \\ 0 & 0 […]
  • The Product of a Subgroup and a Normal Subgroup is a SubgroupThe 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 […]

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