singular matrix

A singular matrix is a square matrix that has no inverse — a broken “undo button.” Where a healthy matrix can be reversed to recover what you started with, a singular one has lost too much information to be reversed at all, like a projection that flattens space onto a line.

The litmus test is the determinant: a square matrix is singular exactly when its determinant equals zero. Equivalently, its rows (or columns) are linearly dependent — at least one is a combination of the others — so the grid is, in a sense, “redundant.” A non-singular matrix, by contrast, has nonzero determinant and does have an inverse.

Singularity is the matrix analogue of the number zero: just as you cannot take 1/0, you cannot invert a singular matrix. For a linear system Ax = b, a singular A means there is no single unique solution — the system either has no solution or infinitely many, never exactly one.