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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