# UFD

• Determine a Matrix From Its Eigenvalue Let $A=\begin{bmatrix} a & -1\\ 1& 4 \end{bmatrix}$ be a $2\times 2$ matrix, where $a$ is some real number. Suppose that the matrix $A$ has an eigenvalue $3$. (a) Determine the value of $a$. (b) Does the matrix $A$ have eigenvalues other than […]
• Finite Group and Subgroup Criteria Let $G$ be a finite group and let $H$ be a subset of $G$ such that for any $a,b \in H$, $ab\in H$. Then show that $H$ is a subgroup of $G$.   Proof. Let $a \in H$. To show that $H$ is a subgroup of $G$, it suffices to show that the inverse $a^{-1}$ is in $H$. If […]
• Eckmann–Hilton Argument: Group Operation is a Group Homomorphism Let $G$ be a group with the identity element $e$ and suppose that we have a group homomorphism $\phi$ from the direct product $G \times G$ to $G$ satisfying $\phi(e, g)=g \text{ and } \phi(g, e)=g, \tag{*}$ for any $g\in G$. Let $\mu: G\times G \to G$ be a map defined […]
• The Possibilities For the Number of Solutions of Systems of Linear Equations that Have More Equations than Unknowns Determine all possibilities for the number of solutions of each of the system of linear equations described below. (a) A system of $5$ equations in $3$ unknowns and it has $x_1=0, x_2=-3, x_3=1$ as a solution. (b) A homogeneous system of $5$ equations in $4$ unknowns and the […]
• A Recursive Relationship for a Power of a Matrix Suppose that the $2 \times 2$ matrix $A$ has eigenvalues $4$ and $-2$. For each integer $n \geq 1$, there are real numbers $b_n , c_n$ which satisfy the relation $A^{n} = b_n A + c_n I ,$ where $I$ is the identity matrix. Find $b_n$ and $c_n$ for $2 \leq n \leq 5$, and […]
• Probability Problems about Two Dice Two fair and distinguishable six-sided dice are rolled. (1) What is the probability that the sum of the upturned faces will equal $5$? (2) What is the probability that the outcome of the second die is strictly greater than the first die? Solution. The sample space $S$ is […]
• 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 […]
• Equivalent Conditions to be a Unitary Matrix A complex matrix is called unitary if $\overline{A}^{\trans} A=I$. The inner product $(\mathbf{x}, \mathbf{y})$ of complex vector $\mathbf{x}$, $\mathbf{y}$ is defined by $(\mathbf{x}, \mathbf{y}):=\overline{\mathbf{x}}^{\trans} \mathbf{y}$. The length of a complex vector […]