(Solved): A matrix \( A \) is given. Determine if the homogeneous system \( A \mathbf{x}=\mathbf{0} \) (wher ...

A matrix \( A \) is given. Determine if the homogeneous system \( A \mathbf{x}=\mathbf{0} \) (where \( \mathbf{x} \) and \( \mathbf{0} \) have the appropriate number of components) has any nontrivial solutions. \[ A=\left[\begin{array}{rrr} -3 & 2 & 1 \\ 1 & -1 & -1 \\ 5 & -6 & -7 \end{array}\right] \] \( A \mathbf{x}=\mathbf{0} \) has nontrivial solutions. \( A \mathbf{x}=\mathbf{0} \) has only the trivial solution.