Linear independence and singular matrices

A set of vectors is linearly independent if no vector in it is a linear combination of the others. A redundant (linearly dependent) vector adds nothing to the span: a 2 × 2 matrix with two identical columns still only reaches a line.

For A⁻¹ to exist, Ax = b must have exactly one solution for every b. That needs A to be square (m = n) with all its columns linearly independent. A square matrix with linearly dependent columns is called singular, and it has no inverse.

A system can have no solution, exactly one, or infinitely many, but never a number in between such as two: if x and y are both solutions, so is αx + (1 − α)y for every real α.