Determine whether the following statements are true or false and provide a justification for your response.
If the coefficient matrix of a system of linear equations has a pivot in the rightmost column, then the system is inconsistent.
If a system of equations has two equations and four unknowns, then it must be consistent.
If a system of equations having four equations and three unknowns is consistent, then the solution is unique.
Suppose that a linear system has four equations and four unknowns and that the coefficient matrix has four pivots. Then the linear system is consistent and has a unique solution.
Suppose that a linear system has five equations and three unknowns and that the coefficient matrix has a pivot in every column. The the linear system is consistent and has a unique solution.