Written Homework 03

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 $n\times n$ matrix. Prove that if $A$ has a column of zeros, then $A$ is not invertible.
  • Hint: there are different ways to approach this; here’s one. Suppose $A$ has a column of zeros. If $A$ is invertible, then there must be an $n\times n$ matrix $B$ such that $BA = I$. Now explain why this is impossible by using a result you proved on the previous written homework.
• Problem 2: Assume $A$ and $B$ are $n \times n$ matrices. Prove that if the product $AB$ is invertible, then $B$ is also invertible.
  • Hint: consider using the portion of the Invertible Matrix Theorem from class that says a matrix $M$ is invertible if and only if $M\bar{x} = \bar{0}$ implies $\bar{x} = \bar{0}$.