# problems in Mathematics

• Compute the Product $A^{2017}\mathbf{u}$ of a Matrix Power and a Vector Let $A=\begin{bmatrix} -1 & 2 \\ 0 & -1 \end{bmatrix} \text{ and } \mathbf{u}=\begin{bmatrix} 1\\ 0 \end{bmatrix}.$ Compute $A^{2017}\mathbf{u}$.   (The Ohio State University, Linear Algebra Exam) Solution. We first compute $A\mathbf{u}$. We […]
• Companion Matrix for a Polynomial Consider a polynomial $p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,$ where $a_i$ are 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 & […]
• Nilpotent Matrix and Eigenvalues of the Matrix An $n\times n$ matrix $A$ is called nilpotent if $A^k=O$, where $O$ is the $n\times n$ zero matrix. Prove the followings. (a) The matrix $A$ is nilpotent if and only if all the eigenvalues of $A$ is zero. (b) The matrix $A$ is nilpotent if and only if […]
• Powers of a Diagonal Matrix Let $A=\begin{bmatrix} a & 0\\ 0& b \end{bmatrix}$. Show that (1) $A^n=\begin{bmatrix} a^n & 0\\ 0& b^n \end{bmatrix}$ for any $n \in \N$. (2) Let $B=S^{-1}AS$, where $S$ be an invertible $2 \times 2$ matrix. Show that $B^n=S^{-1}A^n S$ for any $n \in […] • Is the Sum of a Nilpotent Matrix and an Invertible Matrix Invertible? A square matrix$A$is called nilpotent if some power of$A$is the zero matrix. Namely,$A$is nilpotent if there exists a positive integer$k$such that$A^k=O$, where$O$is the zero matrix. Suppose that$A$is a nilpotent matrix and let$B$be an invertible matrix of […] • 7 Problems on Skew-Symmetric Matrices Let$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 […]