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, prove that $\det(-A) = (-1)^n\det(A)$.

  • Hint: $-A$ is the matrix where every entry of $A$ is multiplied by $-1$. What row operations would you do to transform $A$ to $-A$?

Problem for extra practice

Do not turn in, but I’m happy to talk about it with you if you want.

• Problem: 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: use the previous problem and other properties of the determinant to see what $A^T = -A$ implies about the determinant of $A$.