I'm studying Algebra and I'm asked to prove or disprove "Is the set of all invertible $n \times n$ matrices a vector space?" I assume with respect to the usual matrix-sum and scalar multiplication. I found that is true, but I'm not sure how to prove it.
My problem here is that this statement is too "broad", i.e. I cannot create a matrix with arbitrary values a,b,c,d,(…) considering that I don't know the size of the matrix.
My idea was to prove in a first time that the set of all invertible 2 x 2 matrices of real number is a vector space and then to show that this property could be extended to bigger matrices. However I don't know how to do it and it's precisely here that I need a little help ; how can I show that we can extend our statement?
Is it enough to say that the matrix's size doesn't affect in any way properties we need to check?
Is my overall strategy wrong?
Any form of help would be very appreciated on this dubious answer.
Best Answer
The set of all invertible $n\times n$ matrices of real numbers is NOT a vector space.
Let for example $I$, the unit matrix is invertible and so is $-I$. But their sum $I+(-I)=0$ is definitely not invertible!