If A is invertible then the columns of A to the minus one are linearly independent Explain why

Suppose A is an invertible matrix and let B be its inverse. We want to show that the columns of B are linearly independent.

Suppose for the sake of contradiction that the columns of B are linearly dependent. Then there exist scalars c1, c2, ..., cn, not all zero, such that:

c1B1 + c2B2 + ... + cnBn = 0

where B1, B2, ..., Bn are the columns of B. Multiplying both sides by A on the left, we get:

c1AB1 + c2AB2 + ... + cnABn = 0

But we know that AB = I, the identity matrix, since A is invertible. Therefore, the above equation simplifies to:

c1I + c2I + ... + cnI = 0

or

(c1 + c2 + ... + cn)I = 0

Since c1, c2, ..., cn are not all zero, their sum is not zero. Therefore, we have a nonzero scalar multiple of the identity matrix equal to zero, which is impossible. This contradiction shows that the assumption that the columns of B are linearly dependent must be false, and therefore the columns of B are linearly independent