Final answer:
To show that D is the inverse of an invertible matrix A given that AD=I, we use matrix algebra to multiply both sides by A's inverse to demonstrate that D equals A's inverse.
Step-by-step explanation:
To show that if matrix A is invertible and matrix D satisfies AD=I, where I is the identity matrix, we must demonstrate that D is the inverse of A. If A is invertible, then there exists a matrix A-1 such that A-1A = I and AA-1 = I. Given that AD = I, we can multiply both sides by A-1 to yield A-1(AD) = A-1I.
Using the associative property of matrix multiplication, we can group A-1A together to get (A-1A)D = A-1. Because A-1A equals I, we simplify this to ID = A-1, and since the identity matrix I does not change any matrix it multiplies, D = A-1. Thus, matrix D is the inverse of matrix A, showing that D is in fact A-1.
Therefore, for any invertible matrix A for which AD = I, D must be the inverse of A.