# Prime-Ideal

by Yu ·

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- 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 […]
- If 2 by 2 Matrices Satisfy $A=AB-BA$, then $A^2$ is Zero Matrix Let $A, B$ be complex $2\times 2$ matrices satisfying the relation \[A=AB-BA.\] Prove that $A^2=O$, where $O$ is the $2\times 2$ zero matrix. Hint. Find the trace of $A$. Use the Cayley-Hamilton theorem Proof. We first calculate the […]
- If $A$ is an Idempotent Matrix, then When $I-kA$ is an Idempotent Matrix? A square matrix $A$ is called idempotent if $A^2=A$. (a) Suppose $A$ is an $n \times n$ idempotent matrix and let $I$ be the $n\times n$ identity matrix. Prove that the matrix $I-A$ is an idempotent matrix. (b) Assume that $A$ is an $n\times n$ nonzero idempotent matrix. Then […]
- The Product of a Subgroup and a Normal Subgroup is a Subgroup Let $G$ be a group. Let $H$ be a subgroup of $G$ and let $N$ be a normal subgroup of $G$. The product of $H$ and $N$ is defined to be the subset \[H\cdot N=\{hn\in G\mid h \in H, n\in N\}.\] Prove that the product $H\cdot N$ is a subgroup of […]
- The Ring $\Z[\sqrt{2}]$ is a Euclidean Domain Prove that the ring of integers \[\Z[\sqrt{2}]=\{a+b\sqrt{2} \mid a, b \in \Z\}\] of the field $\Q(\sqrt{2})$ is a Euclidean Domain. Proof. First of all, it is clear that $\Z[\sqrt{2}]$ is an integral domain since it is contained in $\R$. We use the […]
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- Diagonalize a 2 by 2 Matrix if Diagonalizable Determine whether the matrix \[A=\begin{bmatrix} 1 & 4\\ 2 & 3 \end{bmatrix}\] is diagonalizable. If so, find a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$. (The Ohio State University, Linear Algebra Final Exam […]