A useful application of calculating matrix rank is to calculate the number of solutions of linear equations. If the rank of the coefficient matrix is equal to the rank of the augmented matrix, the equations only need one solution. In this case, if its rank is equal to the number of equations, it has an exact solution. If the rank of the augmented matrix is greater than the rank of the coefficient matrix, the general solution has k free parameters, where k is the difference between the number of equations and the rank. Otherwise, the equation is inconsistent.

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