# Math-Magic Tree filled

by Yu ·

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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 chose the numbers $7, 5, 3, 2, […]
- Eigenvalues of Squared Matrix and Upper Triangular Matrix Suppose that $A$ and $P$ are $3 \times 3$ matrices and $P$ is invertible matrix. If \[P^{-1}AP=\begin{bmatrix} 1 & 2 & 3 \\ 0 &4 &5 \\ 0 & 0 & 6 \end{bmatrix},\] then find all the eigenvalues of the matrix $A^2$. We give two proofs. The first version is a […]
- In a Principal Ideal Domain (PID), a Prime Ideal is a Maximal Ideal Let $R$ be a principal ideal domain (PID) and let $P$ be a nonzero prime ideal in $R$. Show that $P$ is a maximal ideal in $R$. Definition A commutative ring $R$ is a principal ideal domain (PID) if $R$ is a domain and any ideal $I$ is generated by a single element […]
- Determine the Dimension of a Mysterious Vector Space From Coordinate Vectors Let $V$ be a vector space and $B$ be a basis for $V$. Let $\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, \mathbf{w}_4, \mathbf{w}_5$ be vectors in $V$. Suppose that $A$ is the matrix whose columns are the coordinate vectors of $\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, […]
- The Matrix $[A_1, \dots, A_{n-1}, A\mathbf{b}]$ is Always Singular, Where $A=[A_1,\dots, A_{n-1}]$ and $\mathbf{b}\in \R^{n-1}$. Let $A$ be an $n\times (n-1)$ matrix and let $\mathbf{b}$ be an $(n-1)$-dimensional vector. Then the product $A\mathbf{b}$ is an $n$-dimensional vector. Set the $n\times n$ matrix $B=[A_1, A_2, \dots, A_{n-1}, A\mathbf{b}]$, where $A_i$ is the $i$-th column vector of […]
- A Simple Abelian Group if and only if the Order is a Prime Number Let $G$ be a group. (Do not assume that $G$ is a finite group.) Prove that $G$ is a simple abelian group if and only if the order of $G$ is a prime number. Definition. A group $G$ is called simple if $G$ is a nontrivial group and the only normal subgroups of $G$ is […]
- Group Homomorphisms From Group of Order 21 to Group of Order 49 Let $G$ be a finite group of order $21$ and let $K$ be a finite group of order $49$. Suppose that $G$ does not have a normal subgroup of order $3$. Then determine all group homomorphisms from $G$ to $K$. Proof. Let $e$ be the identity element of the group […]
- Example of a Nilpotent Matrix $A$ such that $A^2\neq O$ but $A^3=O$. Find a nonzero $3\times 3$ matrix $A$ such that $A^2\neq O$ and $A^3=O$, where $O$ is the $3\times 3$ zero matrix. (Such a matrix is an example of a nilpotent matrix. See the comment after the solution.) Solution. For example, let $A$ be the following $3\times […]