# Linear Algebra Proofs

• Oct 14th 2012, 10:54 AM
tenrear3
Linear Algebra Proofs
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 :)
• Oct 14th 2012, 12:13 PM
Deveno
Re: Linear Algebra Proofs
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:

$\displaystyle A = \begin{bmatrix}1&0\\0&0 \end{bmatrix}; B = \begin{bmatrix}0&0\\0&1 \end{bmatrix}$. 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....
• Oct 14th 2012, 12:35 PM
tenrear3
Re: Linear Algebra Proofs
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.