Linear Independence of Columns in Inverse Matrix: A-1

Suppose A is an invertible matrix. Then, by definition, there exists a matrix A-1 such that AA-1 = I, where I is the identity matrix. Now consider the columns of A-1 denoted by v1, v2, ..., vn. We want to show that these columns are linearly independent.

Suppose there exist scalars c1, c2, ..., cn such that c1v1 + c2v2 + ... + cnvn = 0 We want to show that all ci = 0. Multiplying both sides of the equation by A, we get Ac1v1 + Ac2v2 + ... + Acnvn = 0 Using the fact that AA-1 = I, we can rewrite this as c1(AA-1)v1 + c2(AA-1)v2 + ... + cn(AA-1)vn = 0 Simplifying, we get c1v1 + c2v2 + ... + cnvn = 0 But we already know that c1v1 + c2v2 + ... + cnvn = 0, so we must have c1v1 + c2v2 + ... + cnvn = c1v1 + c2v2 + ... + cnvn Subtracting the two equations, we get 0 = 0 This implies that all ci = 0, which means that the columns of A-1 are linearly independent.