Also please be clear and specific thanksSolutiona No it is

Solution

a)

No it is not a subspace

For a set to be a subspace we require that given any two elements:u,v in the set u+v must also be in the set.

So consider two points in the set:

(-2,1) ,(1,-2)

Adding the two points gives:

(-2,1)+(1,-2)=(-1,-1) which is not in the set.

Hence it is not a subspace.

b)

Three conditions for a set to be a subspace

1. 0 must belong to the subspace

0 matrix ie with all entries 0 belongs to this set.

2. For u,v in the set u+v must also be in the set

Let two matrices:U and V in the set

So, U_{11}=-U_{22}

V_{11}=-V_{22}

Adding two matrices gives:

W=U+V

W_{11}=(U+V)_{11}=U_{11}+V_{11}=-U_{22}-V{22}=-(U+V)_{22}=-W_{22}

Hence, U+V is in the set

3. For any c real number and U in the set, cU must also be in the set

(cU)_{11}=c(U_{11})=-c(U_{22})=-(cU)_{22}

Hence, cU is in the set.

Hence it is a subspace.