A matrix can be undone only if it never squashes space flat
Think of a matrix as a machine that moves every point in space. If it keeps distinct points apart and still reaches everywhere, another matrix can reverse the move exactly. If it flattens space, say squeezing a plane onto a line, information is lost for good, and no inverse can ever restore it.
Formally, a square matrix A is invertible when some matrix B of the same size multiplies it from either side to give the identity matrix, the matrix equivalent of the number one. That partner is unique and is written as A to the power minus one. Applying A to a vector and then its inverse returns the original vector unchanged. Such matrices are also called non-singular, non-degenerate or regular.
A simple test is the determinant, a single number computed from the entries. For real matrices it measures how the transformation scales volume. A nonzero determinant means the matrix is invertible, while zero means volume collapses and no inverse exists. Consider the 2-by-2 matrix whose two rows both read 2 and 4. It sends the vector with entries 2 and minus 1 to zero, just as it sends zero itself to zero. With two different inputs landing on the same output, there is no way to tell which one to return, and its determinant is indeed zero.
Many other conditions turn out to be equivalent, collected in what is called the invertible matrix theorem. The only solution of A times x equals zero is the zero vector; the columns form a basis; the matrix has full rank. Put another way, it is both one-to-one and onto, a perfect pairing of inputs and outputs.
For 2-by-2 matrices there is a neat shortcut: swap the two diagonal entries, flip the signs of the other two, and divide by the determinant. Larger matrices usually call for Gaussian elimination, applying row operations until the matrix becomes the identity. That route has a bonus: if the process ends on something of lower rank instead, it has proved the matrix is not invertible, often faster than computing the determinant.
Source: Invertible matrix
