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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, […]
  • Every Prime Ideal of a Finite Commutative Ring is MaximalEvery Prime Ideal of a Finite Commutative Ring is Maximal Let $R$ be a finite commutative ring with identity $1$. Prove that every prime ideal of $R$ is a maximal ideal of $R$. Proof. We give two proofs. The first proof uses a result of a previous problem. The second proof is self-contained. Proof 1. Let $I$ be a prime ideal […]
  • Rotation Matrix in Space and its Determinant and EigenvaluesRotation Matrix in Space and its Determinant and Eigenvalues For a real number $0\leq \theta \leq \pi$, we define the real $3\times 3$ matrix $A$ by \[A=\begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta &\cos\theta &0 \\ 0 & 0 & 1 \end{bmatrix}.\] (a) Find the determinant of the matrix $A$. (b) Show that $A$ is an […]
  • The Group of Rational Numbers is Not Finitely GeneratedThe Group of Rational Numbers is Not Finitely Generated (a) Prove that the additive group $\Q=(\Q, +)$ of rational numbers is not finitely generated. (b) Prove that the multiplicative group $\Q^*=(\Q\setminus\{0\}, \times)$ of nonzero rational numbers is not finitely generated.   Proof. (a) Prove that the additive […]
  • Sylow Subgroups of a Group of Order 33 is Normal SubgroupsSylow Subgroups of a Group of Order 33 is Normal Subgroups Prove that any $p$-Sylow subgroup of a group $G$ of order $33$ is a normal subgroup of $G$.   Hint. We use Sylow's theorem. Review the basic terminologies and Sylow's theorem. Recall that if there is only one $p$-Sylow subgroup $P$ of $G$ for a fixed prime $p$, then $P$ […]
  • How to Calculate and Simplify a Matrix PolynomialHow to Calculate and Simplify a Matrix Polynomial Let $T=\begin{bmatrix} 1 & 0 & 2 \\ 0 &1 &1 \\ 0 & 0 & 2 \end{bmatrix}$. Calculate and simplify the expression \[-T^3+4T^2+5T-2I,\] where $I$ is the $3\times 3$ identity matrix. (The Ohio State University Linear Algebra Exam) Hint. Use the […]
  • The Polynomial $x^p-2$ is Irreducible Over the Cyclotomic Field of $p$-th Root of UnityThe Polynomial $x^p-2$ is Irreducible Over the Cyclotomic Field of $p$-th Root of Unity Prove that the polynomial $x^p-2$ for a prime number $p$ is irreducible over the field $\Q(\zeta_p)$, where $\zeta_p$ is a primitive $p$th root of unity.   Hint. Consider the field extension $\Q(\sqrt[p]{2}, \zeta)$, where $\zeta$ is a primitive $p$-th root of […]
  • Find Bases for the Null Space, Range, and the Row Space of a $5\times 4$ MatrixFind Bases for the Null Space, Range, and the Row Space of a $5\times 4$ Matrix Let \[A=\begin{bmatrix} 1 & -1 & 0 & 0 \\ 0 &1 & 1 & 1 \\ 1 & -1 & 0 & 0 \\ 0 & 2 & 2 & 2\\ 0 & 0 & 0 & 0 \end{bmatrix}.\] (a) Find a basis for the null space $\calN(A)$. (b) Find a basis of the range $\calR(A)$. (c) Find a basis of the […]

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