Exercise 7

Determine whether the following statements are true or false and explain your reasoning.

1. If $A$ is invertible, then the columns of $A$ are linearly independent.

2. If $A$ is a square matrix whose diagonal entries are all nonzero, then $A$ is invertible.

3. If $A$ is an invertible $n\times n$ matrix, then span of the columns of $A$ is $\real^n$.

4. If $A$ is invertible, then there is a nonzero solution to the homogeneous equation $A\xvec = \zerovec$.

5. If $A$ is an $n\times n$ matrix and the equation $A\xvec = \bvec$ has a solution for every vector $\bvec$, then $A$ is invertible.