If A is invertible then the columns of A-1 are linearly independent Explain why

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