Written Homework 01

Problem List

The problems to turn in are below. Please organize your work, justify your steps, and write in complete sentences with correct punctuation. Once your finish these problems, please follow the directions in the Canvas assignment to submit them. If you have any questions (about the math or writing or submission process or anything), please let me know!

• Problem 1: Let $A$ be an $m\times n$ matrix and $B$ an $n\times p$ matrix. Suppose that some column of $B$ consists entirely of zeros. Prove that some column of the product $AB$ also consists entirely of zeros.
  • Note: please work with arbitrary matrices and an arbitrary column of $B$. Maybe start your proof like this: “Let $A$ be an $m\times n$ matrix and $B$ an $n\times p$ matrix. Suppose that $\operatorname{col}_i(B)$ is the zero vector.” (Now keep going. Try using the definition of matrix multiplication from class.)
• Problem 2: Does there exist a $2\times 2$ matrix $A$ such that the product $A\cdot A^T =\begin{bmatrix} 0 & 1\\ -1 & 0\end{bmatrix}$? Please either give an example of such a matrix $A$ or prove that no such matrix exists.
  • Hint: consider first exploring this for an arbitrary $2\times 2$ matrix $A = \begin{bmatrix} a & b\\ c & d\end{bmatrix}$.