Exercise6

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 the columns of $$A$$ span $$\real^n\text{.}$$

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

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

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