Written Homework 04

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 invertible $n \times n$ matrix. Suppose that $A = A^{-1}$. Prove that $\det(A) = \pm 1$.

• Problem 2: Let $A$ be an $n \times n$ matrix. Using what we learned about how row operations affect the determinant, explain why $\det(-A) = (-1)^n\det(A)$.

• Problem 3: Let $A$ be an $n \times n$ matrix. Suppose that $A^T = -A$. Prove that if $n$ is odd, then $A$ is not invertible.

  • Hint: what does the property $A^T = -A$ imply about the determinant of $A$?