# UFD

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• Given the Data of Eigenvalues, Determine if the Matrix is Invertible In each of the following cases, can we conclude that $A$ is invertible? If so, find an expression for $A^{-1}$ as a linear combination of positive powers of $A$. If $A$ is not invertible, explain why not. (a) The matrix $A$ is a $3 \times 3$ matrix with eigenvalues $\lambda=i , […] • Group of Invertible Matrices Over a Finite Field and its Stabilizer Let$\F_p$be the finite field of$p$elements, where$p$is a prime number. Let$G_n=\GL_n(\F_p)$be the group of$n\times n$invertible matrices with entries in the field$\F_p$. As usual in linear algebra, we may regard the elements of$G_n$as linear transformations on$\F_p^n$, […] • Companion Matrix for a Polynomial Consider a polynomial $p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,$ where$a_iare real numbers. Define the matrix $A=\begin{bmatrix} 0 & 0 & \dots & 0 &-a_0 \\ 1 & 0 & \dots & 0 & -a_1 \\ 0 & 1 & \dots & 0 & -a_2 \\ \vdots & […] • Find a Basis of the Subspace Spanned by Four Polynomials of Degree 3 or Less Let \calP_3 be the vector space of all polynomials of degree 3 or less. Let \[S=\{p_1(x), p_2(x), p_3(x), p_4(x)\},$ where \begin{align*} p_1(x)&=1+3x+2x^2-x^3 & p_2(x)&=x+x^3\\ p_3(x)&=x+x^2-x^3 & p_4(x)&=3+8x+8x^3. \end{align*} (a) […] • Find the Rank of a Matrix with a Parameter Find the rank of the following real matrix. $\begin{bmatrix} a & 1 & 2 \\ 1 &1 &1 \\ -1 & 1 & 1-a \end{bmatrix},$ wherea$is a real number. (Kyoto University, Linear Algebra Exam) Solution. The rank is the number of nonzero rows of a […] • A Subgroup of Index a Prime$p$of a Group of Order$p^n$is Normal Let$G$be a finite group of order$p^n$, where$p$is a prime number and$n$is a positive integer. Suppose that$H$is a subgroup of$G$with index$[G:P]=p$. Then prove that$H$is a normal subgroup of$G$. (Michigan State University, Abstract Algebra Qualifying […] • 7 Problems on Skew-Symmetric Matrices Let$A$and$B$be$n\times n$skew-symmetric matrices. Namely$A^{\trans}=-A$and$B^{\trans}=-B$. (a) Prove that$A+B$is skew-symmetric. (b) Prove that$cA$is skew-symmetric for any scalar$c$. (c) Let$P$be an$m\times n$matrix. Prove that$P^{\trans}AP$is […] • Non-Example of a Subspace in 3-dimensional Vector Space$\R^3$Let$S$be the following subset of the 3-dimensional vector space$\R^3$. $S=\left\{ \mathbf{x}\in \R^3 \quad \middle| \quad \mathbf{x}=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}, x_1, x_2, x_3 \in \Z \right\},$ where$\Z\$ is the set of all integers. […]