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: Assume $A$ and $B$ are row equivalent $n \times n$ matrices. Prove that if A is invertible, then so is $B$.
- Clarification: recall that row equivalent means you can get from one matrix to the other via a series of elementary row operations, which we usually write in class as $A\sim B$.
- 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 Invertibility Theorem from class that says a matrix $M$ is invertibile if and only if the equation $M\bar{x} = \bar{0}$ has only the solution $\bar{x} = \bar{0}$.
- Problem 3: Assume $A$ and $B$ are invertible $n \times n$ matrices. We have seen that $AB$ must also be invertible, but what about $A+B$? Either prove that $A+B$ is invertible (for arbitrary invertible matrices $A$ and $B$), or give an example of invertible matrices $A$ and $B$ where $A+B$ is not invertible.
