The identity matrix and the inverse

The identity matrix Iₙ leaves every n-dimensional vector unchanged: Iₙx = x. It has 1s along the main diagonal and 0s everywhere else.

The inverse of A, written A⁻¹, is the matrix with A⁻¹A = Iₙ. When it exists, it solves Ax = b in one step: multiply both sides by A⁻¹ to get x = A⁻¹b. For square matrices, the left inverse and the right inverse (AA⁻¹ = I) are the same matrix.

Whether A⁻¹ exists at all is the subject of Linear independence and singular matrices.