# Galois-theory-eye-catch

by Yu · Published · Updated

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- Find the Rank of the Matrix $A+I$ if Eigenvalues of $A$ are $1, 2, 3, 4, 5$ Let $A$ be an $n$ by $n$ matrix with entries in complex numbers $\C$. Its only eigenvalues are $1,2,3,4,5$, possibly with multiplicities. What is the rank of the matrix $A+I_n$, where $I_n$ is the identity $n$ by $n$ matrix. (UCB-University of California, Berkeley, […]
- Find a Row-Equivalent Matrix which is in Reduced Row Echelon Form and Determine the Rank For each of the following matrices, find a row-equivalent matrix which is in reduced row echelon form. Then determine the rank of each matrix. (a) $A = \begin{bmatrix} 1 & 3 \\ -2 & 2 \end{bmatrix}$. (b) $B = \begin{bmatrix} 2 & 6 & -2 \\ 3 & -2 & 8 \end{bmatrix}$. (c) $C […]
- Diagonalizable by an Orthogonal Matrix Implies a Symmetric Matrix Let $A$ be an $n\times n$ matrix with real number entries. Show that if $A$ is diagonalizable by an orthogonal matrix, then $A$ is a symmetric matrix. Proof. Suppose that the matrix $A$ is diagonalizable by an orthogonal matrix $Q$. The orthogonality of the […]
- The Inverse Matrix is Unique Let $A$ be an $n\times n$ invertible matrix. Prove that the inverse matrix of $A$ is uniques. Hint. That the inverse matrix of $A$ is unique means that there is only one inverse matrix of $A$. (That's why we say "the" inverse matrix of $A$ and denote it by […]
- The Order of a Conjugacy Class Divides the Order of the Group Let $G$ be a finite group. The centralizer of an element $a$ of $G$ is defined to be \[C_G(a)=\{g\in G \mid ga=ag\}.\] A conjugacy class is a set of the form \[\Cl(a)=\{bab^{-1} \mid b\in G\}\] for some $a\in G$. (a) Prove that the centralizer of an element of $a$ […]
- A One-Line Proof that there are Infinitely Many Prime Numbers Prove that there are infinitely many prime numbers in ONE-LINE. Background There are several proofs of the fact that there are infinitely many prime numbers. Proofs by Euclid and Euler are very popular. In this post, I would like to introduce an elegant one-line […]
- Does the Trace Commute with Matrix Multiplication? Is $\tr (A B) = \tr (A) \tr (B) $? Let $A$ and $B$ be $n \times n$ matrices. Is it always true that $\tr (A B) = \tr (A) \tr (B) $? If it is true, prove it. If not, give a counterexample. Solution. There are many counterexamples. For one, take \[A = \begin{bmatrix} 1 & 0 \\ 0 & 0 […]
- Determine a Condition on $a, b$ so that Vectors are Linearly Dependent Let \[\mathbf{v}_1=\begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix}, \mathbf{v}_2=\begin{bmatrix} 1 \\ a \\ 5 \end{bmatrix}, \mathbf{v}_3=\begin{bmatrix} 0 \\ 4 \\ b \end{bmatrix}\] be vectors in $\R^3$. Determine a […]