(LinearAlgebra) all 2x2 invertible matrices closed under addition?

[Sanglee]

Homework Statement

Suppose V is a vector space.
Is the set of all 2x2 invertible matrices closed under addition? If so, please prove it. If not, please
provide a counter-example.

Homework Equations

The Attempt at a Solution

well i know that what does it mean to be closed under addition. When V is closed under addition, if I suppose vector u and w are in the V, their addition u+w is also in the V, right?

The answer for the question is No.
A counter-example my professor provided is I+(-I)=0
I and (-I) are invertible, but their addition 0 is not invertible. and I know why it's not invertible.
But I don't figure out why it is not closed under addition,,.
If the addition is not invertible, does it mean that the addition is not in the V?

 

[vela]

Homework Statement

Suppose V is a vector space.
Is the set of all 2x2 invertible matrices closed under addition? If so, please prove it. If not, please
provide a counter-example.

Homework Equations

The Attempt at a Solution

well i know that what does it mean to be closed under addition. When V is closed under addition, if I suppose vector u and w are in the V, their addition u+w is also in the V, right?

Yes.

The answer for the question is No.
A counter-example my professor provided is I+(-I)=0
I and (-I) are invertible, but their addition 0 is not invertible. and I know why it's not invertible.
But I don't figure out why it is not closed under addition,,.
If the addition is not invertible, does it mean that the addition is not in the V?

Yes. V consists of only invertible matrices, so 0 is not an element in V. So you have u=I and w=-I are both in V, but their sum u+w=0 is not in V. Therefore V is not closed under addition.

 

[Sanglee]

So clear, easy to understand. Thanks!