Please help

Question

True or false: If AX = B is a system of n linear equations in n unknowns and A^-1 does not exist, then AX = B does not have a unique solution.
 

Please explain and give example

Answer 1

If by "does not have a unique solution" yo mean "either has no solution or has more than one solution", then the answer is"not necessarily true". Indeed, an equation as described has a solution (or solutions) if and only if the rank of the coefficient matrix A is the same as the rank of the augmented matrix (A|B), which is the mtrix formed by adding B as the (n+1)-th column to A. When this condition is satisfied, the equation has a unique solution if and only if A is invertible (i.e. it has an inverse). When A is singular (i.e. it has no inverse or not invertible), this alone does not guarantee the existence of a solution. When the coefficient matrix A and the argmented matrix (A|B) have different ranks, the equation has no solution at ll. An extreme example is when A=0 (zero matrix) and B is not zero. Another example is a system like x+y=1, x+y=2. Of course this one is too simple and obvious. Yet another example is x+y+2z=2, 2x-y+z=1, x-2y-z=1. In these examples, the system is self-contradictory.