inverse of a matrix

The inverse of a matrix is its “undo” partner. For an ordinary number, the inverse of 5 is 1/5 because 5 times 1/5 gives 1. For a matrix A, the inverse is a matrix that multiplies with A to give the identity — the matrix version of getting back to 1.

We write the inverse of A as A with a superscript -1 (A^-1). It is defined by A · A^-1 = I and A^-1 · A = I. Only square matrices can have one, and even then only some do. For a 2-by-2 [a, b; c, d] the inverse is (1 / (ad - bc)) times [d, -b; -c, a], which is valid precisely when ad - bc is not zero.

That denominator reveals everything: a matrix has an inverse exactly when its determinant is nonzero. If the determinant is zero the matrix is singular and no inverse exists — just as you cannot divide an ordinary number by zero. The inverse is the engine behind solving the matrix equation Ax = b as x = A^-1 b.