An Underdetermined Set

Show that the following set does not have a unique solution. How many of the unknowns will be undetermined? Interpret the results given by the left-division method.

2x ~ 4y +Sz = -4
4x – 2y +3z = 4
2x +6y – Sz = 0

• Solution
A MATLAB session to check the ranks looks like
»A = [2,-4,5;-4,-2,3;2,6,-8);
»h = [-4;4;0);
ans =
»rank ([A b))

because the ranks of A and [A b] are equal, a solution exists. However, because the number of unknowns is three, and is one greater than the rank of A, one of the unknowns will be undetermined. An infinite number of solutions exists, and we. can solve for only two of the unknowns in terms of the third unknown. We will obtain these solutions in

example 6.4.-4.
Note that even though the number of equations equals the number of unknowns here, matrix A is singular. (We know this because its rank is less than three.) Thus we cannot the matrix inverse method or Cramer’s method for this problem. If we use the left-division method, MATLAB returns a message warning that the problem is singular, rather than producing an answer.

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