# OSU exam problems in mathematics

• True or False Problems on Midterm Exam 1 at OSU Spring 2018 The following problems are True or False. Let $A$ and $B$ be $n\times n$ matrices. (a) If $AB=B$, then $B$ is the identity matrix. (b) If the coefficient matrix $A$ of the system $A\mathbf{x}=\mathbf{b}$ is invertible, then the system has infinitely many solutions. (c) If $A$ […]
• 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 […]
• Independent Events of Playing Cards A card is chosen randomly from a deck of the standard 52 playing cards. Let $E$ be the event that the selected card is a king and let $F$ be the event that it is a heart. Prove or disprove that the events $E$ and $F$ are independent. Definition of Independence Events […]
• Mathematics About the Number 2018 Happy New Year 2018!! Here are several mathematical facts about the number 2018.   Is 2018 a Prime Number? The number 2018 is an even number, so in particular 2018 is not a prime number. The prime factorization of 2018 is $2018=2\cdot 1009.$ Here $2$ and $1009$ are […]
• 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 […]