Is this True or False ? Justify your answer please.

Question

If A is a 2 x 2 matrix such that AA^T (Transpose) = 0, then A = 0.

Answer 1

First, before creating a proof for this question, it is important to know what the kernel and image of a transformation are. A linear transformation f(x) is a map from one vector space to another. The kernel is the subspace of all vectors in the domain that get sent to 0. The image is the subspace of all vectors in the output space that are f(x) for some x in the input space.

First, it should be clear that det(AA^T)=0, which implies that det(A)=det(A^T)=0.

Therefore, the dimension of the kernel of A is either 1 or 2.

Suppose that the dimension of the kernel of A is 1. Then, since im(A^T)=ker(A)⊥, we know that dim(im(AA^T))=1. However, AA^T=0, which yields a contradiction.

Therefore, the dimension of the kernel of A is 2, which implies that A=0.