A) if A is invertible, then -A is also invertible (with inverse -A-1), but their sum is the 0-matrix, which is not invertible. this means we do NOT have closure under vector addition of invertible matrices.
B) consider the matrices:
. what is A+B?
C) nope. they do have the same dimension, though (look at 3x3 matrices of rank 2).
D)If A and AT have the same number of pivots...then...what? something is missing, here....