# top10mathproblems2017

• Find 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 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 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 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 $f$ […]
• The 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 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 $A^{100}$ 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 […]