top10mathproblems2017

LoadingAdd to solve later

Top 10 Popular Math Problems 2016-2017


LoadingAdd to solve later

Sponsored Links

More from my site

  • Find All Symmetric Matrices satisfying the EquationFind All Symmetric Matrices satisfying the Equation Find all $2\times 2$ symmetric matrices $A$ satisfying $A\begin{bmatrix} 1 \\ -1 \end{bmatrix} = \begin{bmatrix} 2 \\ 3 \end{bmatrix}$? Express your solution using free variable(s).   Solution. Let $A=\begin{bmatrix} a & b\\ c& d \end{bmatrix}$ […]
  • The Rank of the Sum of Two MatricesThe Rank of the Sum of Two Matrices Let $A$ and $B$ be $m\times n$ matrices. Prove that \[\rk(A+B) \leq \rk(A)+\rk(B).\] Proof. Let \[A=[\mathbf{a}_1, \dots, \mathbf{a}_n] \text{ and } B=[\mathbf{b}_1, \dots, \mathbf{b}_n],\] where $\mathbf{a}_i$ and $\mathbf{b}_i$ are column vectors of $A$ and $B$, […]
  • Quiz 7. Find a Basis of the Range, Rank, and Nullity of a MatrixQuiz 7. Find a Basis of the Range, Rank, and Nullity of a Matrix (a) Let $A=\begin{bmatrix} 1 & 3 & 0 & 0 \\ 1 &3 & 1 & 2 \\ 1 & 3 & 1 & 2 \end{bmatrix}$. Find a basis for the range $\calR(A)$ of $A$ that consists of columns of $A$. (b) Find the rank and nullity of the matrix $A$ in part (a).   Solution. (a) […]
  • A Group Homomorphism and an Abelian GroupA Group Homomorphism and an Abelian Group Let $G$ be a group. Define a map $f:G \to G$ by sending each element $g \in G$ to its inverse $g^{-1} \in G$. Show that $G$ is an abelian group if and only if the map $f: G\to G$ is a group homomorphism.   Proof. $(\implies)$ If $G$ is an abelian group, then $f$ […]
  • Determine a 2-Digit Number Satisfying Two ConditionsDetermine a 2-Digit Number Satisfying Two Conditions A 2-digit number has two properties: The digits sum to 11, and if the number is written with digits reversed, and subtracted from the original number, the result is 45. Find the number.   Solution. The key to this problem is noticing that our 2-digit number can be […]
  • The Inner Product on $\R^2$ induced by a Positive Definite Matrix and Gram-Schmidt OrthogonalizationThe Inner Product on $\R^2$ induced by a Positive Definite Matrix and Gram-Schmidt Orthogonalization Consider the $2\times 2$ real matrix \[A=\begin{bmatrix} 1 & 1\\ 1& 3 \end{bmatrix}.\] (a) Prove that the matrix $A$ is positive definite. (b) Since $A$ is positive definite by part (a), the formula \[\langle \mathbf{x}, […]
  • All the Conjugacy Classes of the Dihedral Group $D_8$ of Order 8All the Conjugacy Classes of the Dihedral Group $D_8$ of Order 8 Determine all the conjugacy classes of the dihedral group \[D_{8}=\langle r,s \mid r^4=s^2=1, sr=r^{-1}s\rangle\] of order $8$. Hint. You may directly compute the conjugates of each element but we are going to use the following theorem to simplify the […]
  • Use the Cayley-Hamilton Theorem to Compute the Power $A^{100}$Use the Cayley-Hamilton Theorem to Compute the Power $A^{100}$ Let $A$ be a $3\times 3$ real orthogonal matrix with $\det(A)=1$. (a) If $\frac{-1+\sqrt{3}i}{2}$ is one of the eigenvalues of $A$, then find the all the eigenvalues of $A$. (b) Let \[A^{100}=aA^2+bA+cI,\] where $I$ is the $3\times 3$ identity matrix. Using the […]

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.