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Prove or give a counter example of the following
A)The set of all n x n invertible matrices is a subspace
B)The set of all n x n non invertible is a subspace.
C) Ifm= n then the row space of A is equal to the colomn space
D) If A and AT have the same number of pivots
I'm new here so thanks :)
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....
D) Does A and A*T have the same number of Pivots I think. Any other answers to these problems to expand what the person above said.