True or false If Ax 0 has an infinite number of solutions t

True or false. If Ax = 0 has an infinite number of solutions then every system Ax = b is consistent. If true give a proof, otherwise give a counterexample.

Solution

Infinite solution of Ax=0 occurs when number of equations are less than number of variables.

same will be true for Ax=b because after adding non zero constant, number of equations will still be less than number of variables.

Hence solution exists for Ax=b

which means Ax=b is consistent is true.