# 2568syllabus_f16

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- Two Matrices with the Same Characteristic Polynomial. Diagonalize if Possible. Let \[A=\begin{bmatrix} 1 & 3 & 3 \\ -3 &-5 &-3 \\ 3 & 3 & 1 \end{bmatrix} \text{ and } B=\begin{bmatrix} 2 & 4 & 3 \\ -4 &-6 &-3 \\ 3 & 3 & 1 \end{bmatrix}.\] For this problem, you may use the fact that both matrices have the same characteristic […]
- The Product of a Subgroup and a Normal Subgroup is a Subgroup Let $G$ be a group. Let $H$ be a subgroup of $G$ and let $N$ be a normal subgroup of $G$. The product of $H$ and $N$ is defined to be the subset \[H\cdot N=\{hn\in G\mid h \in H, n\in N\}.\] Prove that the product $H\cdot N$ is a subgroup of […]
- For Which Choices of $x$ is the Given Matrix Invertible? Determine the values of $x$ so that the matrix \[A=\begin{bmatrix} 1 & 1 & x \\ 1 &x &x \\ x & x & x \end{bmatrix}\] is invertible. For those values of $x$, find the inverse matrix $A^{-1}$. Solution. We use the fact that a matrix is invertible […]
- Group Homomorphism Sends the Inverse Element to the Inverse Element Let $G, G'$ be groups. Let $\phi:G\to G'$ be a group homomorphism. Then prove that for any element $g\in G$, we have \[\phi(g^{-1})=\phi(g)^{-1}.\] Definition (Group homomorphism). A map $\phi:G\to G'$ is called a group homomorphism […]
- The Product of Two Nonsingular Matrices is Nonsingular Prove that if $n\times n$ matrices $A$ and $B$ are nonsingular, then the product $AB$ is also a nonsingular matrix. (The Ohio State University, Linear Algebra Final Exam Problem) Definition (Nonsingular Matrix) An $n\times n$ matrix is called nonsingular if the […]
- Two Eigenvectors Corresponding to Distinct Eigenvalues are Linearly Independent Let $A$ be an $n\times n$ matrix. Suppose that $\lambda_1, \lambda_2$ are distinct eigenvalues of the matrix $A$ and let $\mathbf{v}_1, \mathbf{v}_2$ be eigenvectors corresponding to $\lambda_1, \lambda_2$, respectively. Show that the vectors $\mathbf{v}_1, \mathbf{v}_2$ are […]
- Possibilities For the Number of Solutions for a Linear System Determine whether the following systems of equations (or matrix equations) described below has no solution, one unique solution or infinitely many solutions and justify your answer. (a) \[\left\{ \begin{array}{c} ax+by=c \\ dx+ey=f, \end{array} \right. \] where $a,b,c, d$ […]
- If matrix product $AB$ is a square, then is $BA$ a square matrix? Let $A$ and $B$ are matrices such that the matrix product $AB$ is defined and $AB$ is a square matrix. Is it true that the matrix product $BA$ is also defined and $BA$ is a square matrix? If it is true, then prove it. If not, find a […]