# Math-Magic Tree Trick

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• The Trick of a Mathematical Game. The One’s Digit of the Sum of Two Numbers. Decipher the trick of the following mathematical magic.   The Rule of the Game Here is the game. Pick six natural numbers ($1, 2, 3, \dots$) and place them in the yellow discs of the picture below. For example, let's say I have chosen the numbers $7, 5, 3, 2, […] • Generators of the Augmentation Ideal in a Group Ring Let$R$be a commutative ring with$1$and let$G$be a finite group with identity element$e$. Let$RG$be the group ring. Then the map$\epsilon: RG \to R$defined by $\epsilon(\sum_{i=1}^na_i g_i)=\sum_{i=1}^na_i,$ where$a_i\in R$and$G=\{g_i\}_{i=1}^n$, is a ring […] • Special Linear Group is a Normal Subgroup of General Linear Group Let$G=\GL(n, \R)$be the general linear group of degree$n$, that is, the group of all$n\times n$invertible matrices. Consider the subset of$G$defined by $\SL(n, \R)=\{X\in \GL(n,\R) \mid \det(X)=1\}.$ Prove that$\SL(n, \R)$is a subgroup of$G$. Furthermore, prove that […] • Given the Characteristic Polynomial, Find the Rank of the Matrix Let$A$be a square matrix and its characteristic polynomial is give by $p(t)=(t-1)^3(t-2)^2(t-3)^4(t-4).$ Find the rank of$A$. (The Ohio State University, Linear Algebra Final Exam Problem) Solution. Note that the degree of the characteristic polynomial […] • A Symmetric Positive Definite Matrix and An Inner Product on a Vector Space (a) Suppose that$A$is an$n\times n$real symmetric positive definite matrix. Prove that $\langle \mathbf{x}, \mathbf{y}\rangle:=\mathbf{x}^{\trans}A\mathbf{y}$ defines an inner product on the vector space$\R^n$. (b) Let$A$be an$n\times n$real matrix. Suppose […] • No Finite Abelian Group is Divisible A nontrivial abelian group$A$is called divisible if for each element$a\in A$and each nonzero integer$k$, there is an element$x \in A$such that$x^k=a$. (Here the group operation of$A$is written multiplicatively. In additive notation, the equation is written as$kx=a$.) That […] • Possibilities For the Number of Solutions for a Linear System Determine whether the following systems of equations (or matrix equations) described below has no solution, one unique solution or infinitely many solutions and justify your answer. (a) $\left\{ \begin{array}{c} ax+by=c \\ dx+ey=f, \end{array} \right.$ where$a,b,c, d$[…] • Abelian Normal subgroup, Quotient Group, and Automorphism Group Let$G$be a finite group and let$N$be a normal abelian subgroup of$G$. Let$\Aut(N)$be the group of automorphisms of$G$. Suppose that the orders of groups$G/N$and$\Aut(N)$are relatively prime. Then prove that$N\$ is contained in the center of […]