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  • The Trick of a Mathematical Game. The One’s Digit of the Sum of Two Numbers.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, […]
  • Isomorphism Criterion of Semidirect Product of GroupsIsomorphism Criterion of Semidirect Product of Groups Let $A$, $B$ be groups. Let $\phi:B \to \Aut(A)$ be a group homomorphism. The semidirect product $A \rtimes_{\phi} B$ with respect to $\phi$ is a group whose underlying set is $A \times B$ with group operation \[(a_1, b_1)\cdot (a_2, b_2)=(a_1\phi(b_1)(a_2), b_1b_2),\] where $a_i […]
  • Find Inverse Matrices Using Adjoint MatricesFind Inverse Matrices Using Adjoint Matrices Let $A$ be an $n\times n$ matrix. The $(i, j)$ cofactor $C_{ij}$ of $A$ is defined to be \[C_{ij}=(-1)^{ij}\det(M_{ij}),\] where $M_{ij}$ is the $(i,j)$ minor matrix obtained from $A$ removing the $i$-th row and $j$-th column. Then consider the $n\times n$ matrix […]
  • The Center of the Heisenberg Group Over a Field $F$ is Isomorphic to the Additive Group $F$The Center of the Heisenberg Group Over a Field $F$ is Isomorphic to the Additive Group $F$ Let $F$ be a field and let \[H(F)=\left\{\, \begin{bmatrix} 1 & a & b \\ 0 &1 &c \\ 0 & 0 & 1 \end{bmatrix} \quad \middle| \quad \text{ for any} a,b,c\in F\, \right\}\] be the Heisenberg group over $F$. (The group operation of the Heisenberg group is matrix […]
  • Quiz 13 (Part 1) Diagonalize a MatrixQuiz 13 (Part 1) Diagonalize a Matrix Let \[A=\begin{bmatrix} 2 & -1 & -1 \\ -1 &2 &-1 \\ -1 & -1 & 2 \end{bmatrix}.\] Determine whether the matrix $A$ is diagonalizable. If it is diagonalizable, then diagonalize $A$. That is, find a nonsingular matrix $S$ and a diagonal matrix $D$ such that […]
  • Determine a Matrix From Its EigenvalueDetermine 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 […]
  • If a Finite Group Acts on a Set Freely and Transitively, then the Numbers of Elements are the SameIf a Finite Group Acts on a Set Freely and Transitively, then the Numbers of Elements are the Same Let $G$ be a finite group and let $S$ be a non-empty set. Suppose that $G$ acts on $S$ freely and transitively. Prove that $|G|=|S|$. That is, the number of elements in $G$ and $S$ are the same.   Definition (Free and Transitive Group Action) A group action of […]
  • Mathematics About the Number 2017Mathematics About the Number 2017 Happy New Year 2017!! Here is the list of mathematical facts about the number 2017 that you can brag about to your friends or family as a math geek. 2017 is a prime number Of course, I start with the fact that the number 2017 is a prime number. The previous prime year was […]

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