Question

if Null(A^T) != {0}, then A is not invertible?
Where A is an mxn matrix.
(!= means not equal to)

Answer 1

If A is a mxn matrix and m != n then it can't be invertible. Only square matrices are. It can only have a right or a left inverse.

Answer 2

Btw what's the T there in Null(A^T) ?

Answer 3

A^T means Transpose of A

Answer 4

"if Null(A^T) != {0}, then A is not invertible?"

Right. Let's say A is an nxn matrix so A^T is nxn as well. Since A^T has non-zero vectors in its nullspace it can't be full rank. Remember rankB = n - dim(N(B)). Since A and A^T have the same rank A isn't full rank either. (rankA < n). Therefore it's not invertible.

The last step can be understood through the rank formula as well. Since:

r < n
r = n - dim (N(A))
so

dim (N(A)) > 0, therefore Ax = 0 for some nonzero vector and A is not invertible.

Answer 5

for invertible the given matrix must be square so it is not invertible