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Find Eigenvalues and Eigenvectors. MIT Linear Algebra homework problem and solution


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  • Group of Order $pq$ Has a Normal Sylow Subgroup and SolvableGroup of Order $pq$ Has a Normal Sylow Subgroup and Solvable Let $p, q$ be prime numbers such that $p>q$. If a group $G$ has order $pq$, then show the followings. (a) The group $G$ has a normal Sylow $p$-subgroup. (b) The group $G$ is solvable.   Definition/Hint For (a), apply Sylow's theorem. To review Sylow's theorem, […]
  • Pick Two Balls from a Box, What is the Probability Both are Red?Pick Two Balls from a Box, What is the Probability Both are Red? There are three blue balls and two red balls in a box. When we randomly pick two balls out of the box without replacement, what is the probability that both of the balls are red? Solution. Let $R_1$ be the event that the first ball is red and $R_2$ be the event that the […]
  • In a Field of Positive Characteristic, $A^p=I$ Does Not Imply that $A$ is Diagonalizable.In a Field of Positive Characteristic, $A^p=I$ Does Not Imply that $A$ is Diagonalizable. Show that the matrix $A=\begin{bmatrix} 1 & \alpha\\ 0& 1 \end{bmatrix}$, where $\alpha$ is an element of a field $F$ of characteristic $p>0$ satisfies $A^p=I$ and the matrix is not diagonalizable over $F$ if $\alpha \neq 0$. Comment. Remark that if $A$ is a square […]
  • Find a Basis for a Subspace of the Vector Space of $2\times 2$ MatricesFind a Basis for a Subspace of the Vector Space of $2\times 2$ Matrices Let $V$ be the vector space of all $2\times 2$ matrices, and let the subset $S$ of $V$ be defined by $S=\{A_1, A_2, A_3, A_4\}$, where \begin{align*} A_1=\begin{bmatrix} 1 & 2 \\ -1 & 3 \end{bmatrix}, \quad A_2=\begin{bmatrix} 0 & -1 \\ 1 & 4 […]
  • Dihedral Group and Rotation of the PlaneDihedral Group and Rotation of the Plane Let $n$ be a positive integer. Let $D_{2n}$ be the dihedral group of order $2n$. Using the generators and the relations, the dihedral group $D_{2n}$ is given by \[D_{2n}=\langle r,s \mid r^n=s^2=1, sr=r^{-1}s\rangle.\] Put $\theta=2 \pi/n$. (a) Prove that the matrix […]
  • The Number of Elements in a Finite Field is a Power of a Prime NumberThe Number of Elements in a Finite Field is a Power of a Prime Number Let $\F$ be a finite field of characteristic $p$. Prove that the number of elements of $\F$ is $p^n$ for some positive integer $n$. Proof. First note that since $\F$ is a finite field, the characteristic of $\F$ must be a prime number $p$. Then $\F$ contains the […]
  • Find the Formula for the Power of a Matrix Using Linear Recurrence RelationFind the Formula for the Power of a Matrix Using Linear Recurrence Relation Suppose that $A$ is $2\times 2$ matrix that has eigenvalues $-1$ and $3$. Then for each positive integer $n$ find $a_n$ and $b_n$ such that \[A^{n+1}=a_nA+b_nI,\] where $I$ is the $2\times 2$ identity matrix.   Solution. Since $-1, 3$ are eigenvalues of the […]
  • If $A^{\trans}A=A$, then $A$ is a Symmetric Idempotent MatrixIf $A^{\trans}A=A$, then $A$ is a Symmetric Idempotent Matrix Let $A$ be a square matrix such that \[A^{\trans}A=A,\] where $A^{\trans}$ is the transpose matrix of $A$. Prove that $A$ is idempotent, that is, $A^2=A$. Also, prove that $A$ is a symmetric matrix.     Hint. Recall the basic properties of transpose […]

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