Prove: if A matrix is invertible then -A matrix is invertible too ?

Answer 1

If A is in invertible, then there is a matrix A^-1 such that AA^-1=A^-1A=I. So consider -A^-1.

(-A)(-A^-1)=(-1)(-1)AA^-1=AA^-1=I, and (-A^-1)(-A)=(-1)(-1)A^-1A=A^-1A=I.

So, we have shown that (-A)^-1=-A^-1. Therefore, -A is invertible.