# module-theory-eye-catch

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- Is there an Odd Matrix Whose Square is $-I$? Let $n$ be an odd positive integer. Determine whether there exists an $n \times n$ real matrix $A$ such that \[A^2+I=O,\] where $I$ is the $n \times n$ identity matrix and $O$ is the $n \times n$ zero matrix. If such a matrix $A$ exists, find an example. If not, prove that […]
- The 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 […]
- Show the Subset of the Vector Space of Polynomials is a Subspace and Find its Basis Let $P_3$ be the vector space over $\R$ of all degree three or less polynomial with real number coefficient. Let $W$ be the following subset of $P_3$. \[W=\{p(x) \in P_3 \mid p'(-1)=0 \text{ and } p^{\prime\prime}(1)=0\}.\] Here $p'(x)$ is the first derivative of $p(x)$ and […]
- A Matrix Commuting With a Diagonal Matrix with Distinct Entries is Diagonal Let \[D=\begin{bmatrix} d_1 & 0 & \dots & 0 \\ 0 &d_2 & \dots & 0 \\ \vdots & & \ddots & \vdots \\ 0 & 0 & \dots & d_n \end{bmatrix}\] be a diagonal matrix with distinct diagonal entries: $d_i\neq d_j$ if $i\neq j$. Let $A=(a_{ij})$ be an $n\times n$ matrix […]
- Determine Whether the Following Matrix Invertible. If So Find Its Inverse Matrix. Let A be the matrix \[\begin{bmatrix} 1 & -1 & 0 \\ 0 &1 &-1 \\ 0 & 0 & 1 \end{bmatrix}.\] Is the matrix $A$ invertible? If not, then explain why it isn’t invertible. If so, then find the inverse. (The Ohio State University Linear Algebra […]
- Powers of a Matrix Cannot be a Basis of the Vector Space of Matrices Let $n>1$ be a positive integer. Let $V=M_{n\times n}(\C)$ be the vector space over the complex numbers $\C$ consisting of all complex $n\times n$ matrices. The dimension of $V$ is $n^2$. Let $A \in V$ and consider the set \[S_A=\{I=A^0, A, A^2, \dots, A^{n^2-1}\}\] of $n^2$ […]
- The Existence of an Element in an Abelian Group of Order the Least Common Multiple of Two Elements Let $G$ be an abelian group. Let $a$ and $b$ be elements in $G$ of order $m$ and $n$, respectively. Prove that there exists an element $c$ in $G$ such that the order of $c$ is the least common multiple of $m$ and $n$. Also determine whether the statement is true if $G$ is a […]
- Every Group of Order 72 is Not a Simple Group Prove that every finite group of order $72$ is not a simple group. Definition. A group $G$ is said to be simple if the only normal subgroups of $G$ are the trivial group $\{e\}$ or $G$ itself. Hint. Let $G$ be a group of order $72$. Use the Sylow's theorem and determine […]