Linear Algebra 29 | Identity and Inverses

Identity matrices and inverses sit at the center of linear algebra because they formalize “do nothing” and “undo what a transformation does.” An n×n identity matrix has 1s on the diagonal and 0s everywhere else, and it acts as the neutral element for matrix multiplication: multiplying any compatible matrix B by the identity leaves B unchanged. On the linear-map level, the identity matrix corresponds to the identity map on R^n, sending every vector x back to itself. These two viewpoints—matrix multiplication and linear transformations—stay tightly linked and provide the same underlying behavior.

Once that neutral element is in place, inverses become the natural next question: what matrix (or map) reverses an operation? For a square matrix A, an inverse is another square matrix A^{-1} of the same size such that A·A^{-1} equals the identity matrix and also A^{-1}·A equals the identity matrix. The requirement that both multiplication orders produce the identity mirrors the familiar real-number idea, but matrices are more restrictive: not every square matrix has such an inverse. If an inverse exists, it is unique—so the notation A^{-1} is justified. Matrices that do have inverses are called invertible; they are also described as non-singular or regular. Matrices without an inverse are called singular (or non-invertible/non-regular).

The same classification can be expressed using the associated linear map f_A. A matrix A is invertible exactly when the induced linear map f_A: R^n → R^n is bijective, meaning it both hits every vector in the codomain and never collapses distinct inputs to the same output. In that case, the inverse map f_A^{-1} exists, and it corresponds to the inverse matrix A^{-1}. This correspondence is not just conceptual—it yields concrete equalities: composing f_A^{-1} with f_A gives the identity map on R^n, and composing in the other order also gives the identity map. Translating back to matrices, those inverse-map relationships match the two matrix equations A·A^{-1} = I and A^{-1}·A = I.

In short, identity matrices define the “do nothing” baseline for both multiplication and linear transformations. Inverses then define the “undo” operation, but only for matrices whose action is bijective at the map level. That equivalence between inverse matrices and inverse linear maps is the key bridge that makes later computations possible—because it turns the abstract idea of reversing a transformation into precise algebraic conditions on matrices.