Exercise 8

Provide a justification for your response to the following questions.

1. Suppose that $A$ is a square matrix with two identical columns. Can $A$ be invertible?

2. Suppose that $A$ is a square matrix with two identical rows. Can $A$ be invertible?

3. Suppose that $A$ is an invertible matrix and that $AB = AC$. Can you conclude that $B = C$?

4. Suppose that $A$ is an invertible $n\times n$ matrix. What can you say about the span of the columns of $A^{-1}$?

5. Suppose that $A$ is an invertible matrix and that $B$ is row equivalent to $A$. Can you guarantee that $B$ is invertible?