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 $n\times n$ matrix. Prove that if $A$ has a column of zeros, then $A$ is not invertible.
  • Hint: Suppose such a matrix were 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: Let $A$ be an $n\times n$ matrix, and suppose that $A^2 + A - I = 0$ where $I$ is the $n\times n$ identity matrix and $0$ is the $n\times n$ zero matrix.
  1. Prove that $A$ is invertible.
  2. Prove that $B^2 - B - I = 0$ for $B = A^{-1}$.
     • Hint: For the first part, try to manipulate $A^2 + A - I = 0$ into the form $AB = I$ for some matrix $B$, and then also show that $BA = I$. This will show that $A$ is invertible and $B$ is $A^{-1}$. You should have a concrete expresion for $B$, which you can then use in the second part.