Written Homework 02

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 the $i^\text{th}$ column of $B$ consists entirely of zeros. Prove that the $i^\text{th}$ column of the product $AB$ also consists entirely of zeros.
  • Note: Please work with arbitrary matrices. Try using the definition of matrix multiplication from class where $AB$ is defined column by column. Your proof may be quite short, but make sure to clearly explain your reasoning.
• 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 exploring this for an arbitrary $2\times 2$ matrix $A = \begin{bmatrix} a & b\\ c & d\end{bmatrix}$.