Let A be a real nxn matrix Prove that AtA is invertible Let

Let A be a real nxn matrix. Prove that AtA is invertible

Let a be a real n Times n matrix. Prove that A^T A is invertible iff the column vectors of A are linearly independent.

Solution

i have some idea regarding this question

We know that the columns of A are linearly independent if and only if Ax = 0 has only the trivial solution. So suppose that Ax = 0. Then we need to prove that x = 0. Of course, ATAx = AT0 = 0, so x Nul(ATA). But ATA is invertible, so by the IMT, the only vector in its null space is 0. This implies that x = 0 as required.

C = CIn = C(AB) = (CA)B = InB = B.