Suppose a system has infinitely many solutions What must be

Suppose a system has infinitely many solutions. What must be true of the number of pivots in the reduced matrix of the system? why? Solution:

Solution

Let A be the augmented matrix of a consistent system of linear equations with n variables. Suppose also that B is a row-equivalent matrix in reduced row-echelon form with r pivot columns. Then r n. If r = n , then the system has a unique solution, and if r < n, then the system has infinitely many solutions.

B has n + 1 columns, so there can be at most n + 1 pivot columns, i.e. r n + 1. If r = n + 1, then every column of B is a pivot column, and in particular, the last column is a pivot column. So the system is inconsistent . We are left with r n.

When r = n, we find n r = 0 free variables and the only solution is given by setting the n variables to the the first n entries of column n + 1 of B.

When r < n, and n r > 0 free variables. Choose one free variable and set all the other free variables to zero. Now, set the chosen free variable to any fixed value. It is possible to then determine the values of the dependent variables to create a solution to the system. By setting the chosen free variable to different values, in this manner infinitely many solutions can be created