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.