When might the linear mapping Ax=b have (a) no solution, (b) a unique solution, (c) multiple solutions?

A is a matrix.

When might the linear mapping Ax=b have (a) no solution, (b) a unique solution, (c) multiple solutions?
Since you are asking this question, I assume that you are familiar with the row reduction algorithm. If you aren't, then you will probably need to study some linear algebra to understand my answer. (I don't know of a way to answer this question without using some basic terms from linear algebra.)





To answer your question, we need to look at the pivot columns of the matrix [A | b], where by [A | b] I mean the matrix we get when we add b as an extra column on the right of A.





A "pivot column" in [A | b] is any column which contains a leading 1 in the row-reduced echelon form of [A | b]. (By a "leading 1" I mean an entry "1" in the matrix such that each entry to the left of the 1 in that row is 0.)





(a) If the rightmost column of [A | b] is a pivot column, then the equation Ax = b has no solution.





(b) If there are as many pivot columns in [A | b] as there are rows in [A | b], but the rightmost column of [A | b] is not a pivot column, then the equation Ax = b has a unique solution.





(c) Otherwise, the equation Ax = b has infinitely many solutions.