The Inverse Matrix

Linear Algebra (Matrices) · Axiom Academy

The inverse A^ -1 is the one matrix that undoes A : the unique solution of AA^ -1 =A^ -1 A=I , and it exists only when . 1. The Inverse Is What Undoes A Think of A as a transformation that reshapes the grid — it sends the basis vectors to the columns of A , bending the unit square into a slanted parallelogram. The inverse A^ -1 is the transformation that takes that reshaped grid and snaps it exactly back to the identity grid . Doing A then A^ -1 leaves every point where it began — that round trip is the definition. A^ -1 snaps it back to I (undo) 2. When Does an Inverse Exist? Not every square matrix can be undone. The deciding quantity is the determinant — the signed area the unit square covers after the transform. If A keeps that area nonzero, no information is lost and the grid can be snapped back. But if , the transform collapses space onto a line : the two basis vectors land on top of the same direction, the area is crushed to zero, and infinitely many starting points get squashed to the same place. Nothing can pull them apart again, so no inverse can exist. Area stays nonzero, the map is reversible, and A^ -1 exists. Such an A is nonsingular . Area collapses to 0 , the map flattens space, and no A^ -1 exists. Such an A is singular . Only a square ( ) matrix can have an inverse — both AA^ -1 and A^ -1 A must equal the same I . A collapse sends many inputs to one output, so the step can't be reversed — there is no single "undo" map. 3. Unique — and Its Own Inverse Is A

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