Finding Inverse Transformations

Linear Algebra (Matrices) · Axiom Academy

EXAMPLE Finding the Inverse Transformation T^ -1 Given an invertible linear transformation , find T^ -1 and confirm it undoes T . The linear transformation is defined by , where . Find the inverse transformation T^ -1 (whose matrix is A^ -1 ), and verify that . The inverse undoes the transformation T carries to ; the inverse transformation T^ -1 carries straight back to by multiplying by A^ -1 . Nice work! You found the inverse of a linear transformation and confirmed it undoes the original. Here's what carried the solution: Invertibility test: is invertible exactly when . Here , so T^ -1 exists. The inverse formula: for , . The matrix of T^ -1 is A^ -1 . Always verify: A^ -1 A = I , so for every — the round-trip returns the input. Geometric meaning: T^ -1 undoes T . If T shears and stretches the plane, T^ -1 applies exactly the reverse so each point lands back where it started. The same logic scales to larger systems — solving for is exactly applying T^ -1 , which is why invertible transformations are the ones you can "run backwards."

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.