# Nagoya-university-exam-eye-catch

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• Find All the Eigenvalues of 4 by 4 Matrix Find all the eigenvalues of the matrix $A=\begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 &0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{bmatrix}.$ (The Ohio State University, Linear Algebra Final Exam Problem)   Solution. We compute the […]
• Every Maximal Ideal of a Commutative Ring is a Prime Ideal Let $R$ be a commutative ring with unity. Then show that every maximal ideal of $R$ is a prime ideal.   We give two proofs. Proof 1. The first proof uses the following facts. Fact 1. An ideal $I$ of $R$ is a prime ideal if and only if $R/I$ is an integral […]
• 10 True of False Problems about Nonsingular / Invertible Matrices 10 questions about nonsingular matrices, invertible matrices, and linearly independent vectors. The quiz is designed to test your understanding of the basic properties of these topics. You can take the quiz as many times as you like. The solutions will be given after […]
• Express the Eigenvalues of a 2 by 2 Matrix in Terms of the Trace and Determinant Let $A=\begin{bmatrix} a & b\\ c& d \end{bmatrix}$ be an $2\times 2$ matrix. Express the eigenvalues of $A$ in terms of the trace and the determinant of $A$.   Solution. Recall the definitions of the trace and determinant of $A$: $\tr(A)=a+d \text{ and } […] • Elementary Questions about a Matrix Let \[A=\begin{bmatrix} -5 & 0 & 1 & 2 \\ 3 &8 & -3 & 7 \\ 0 & 11 & 13 & 28 \end{bmatrix}.$ (a) What is the size of the matrix $A$? (b) What is the third column of $A$? (c) Let $a_{ij}$ be the $(i,j)$-entry of $A$. Calculate $a_{23}-a_{31}$. […]
• 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 […]
• Determine Null Spaces of Two Matrices Let $A=\begin{bmatrix} 1 & 2 & 2 \\ 2 &3 &2 \\ -1 & -3 & -4 \end{bmatrix} \text{ and } B=\begin{bmatrix} 1 & 2 & 2 \\ 2 &3 &2 \\ 5 & 3 & 3 \end{bmatrix}.$ Determine the null spaces of matrices $A$ and $B$.   Proof. The null space of the […]
• Compute and Simplify the Matrix Expression Including Transpose and Inverse Matrices Let $A, B, C$ be the following $3\times 3$ matrices. \[A=\begin{bmatrix} 1 & 2 & 3 \\ 4 &5 &6 \\ 7 & 8 & 9 \end{bmatrix}, B=\begin{bmatrix} 1 & 0 & 1 \\ 0 &3 &0 \\ 1 & 0 & 5 \end{bmatrix}, C=\begin{bmatrix} -1 & 0\ & 1 \\ 0 &5 &6 \\ 3 & 0 & […]