# OSU exam problems in mathematics

• Determine Whether Given Subsets in $\R^4$ are Subspaces or Not (a) Let $S$ be the subset of $\R^4$ consisting of vectors $\begin{bmatrix} x \\ y \\ z \\ w \end{bmatrix}$ satisfying $2x+4y+3z+7w+1=0.$ Determine whether $S$ is a subspace of $\R^4$. If so prove it. If not, explain why it is not a […]
• Find all Values of x such that the Given Matrix is Invertible Let $A=\begin{bmatrix} 2 & 0 & 10 \\ 0 &7+x &-3 \\ 0 & 4 & x \end{bmatrix}.$ Find all values of $x$ such that $A$ is invertible. (Stanford University Linear Algebra Exam) Hint. Calculate the determinant of the matrix $A$. Solution. A […]
• 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 […]