# Answer to Question #7824 in Abstract Algebra for Vais

Question #7824

Linear mathematics Question

The set M2,2 of 2 × 2 matrices, with real entries, is a vector space.

The set of antisymmetric matrices A = { 0 a }

−a 0

where a belongs to R is a subset of M2,2.

Prove that A is a subspace of M2,2.

The set M2,2 of 2 × 2 matrices, with real entries, is a vector space.

The set of antisymmetric matrices A = { 0 a }

−a 0

where a belongs to R is a subset of M2,2.

Prove that A is a subspace of M2,2.

Expert's answer

We have to show that if matrices

P = [ [0,p]; [-p,0] ]

and

Q = [

[0,q]; [-q,0] ]

belong to A, and t is a real number, then

P+Q, and tP

belong to A as well.

Notice that

P+Q = [ [0,p+q]; [-p-q,0] ] = [

[0,p+q]; [-(p+q),0] ]

and

tP = [ [0,tp]; [-tp,0] ]

Both matrices

are antisymmetric, so they belong to A,

and therefore A is a subspace of M2,2

P = [ [0,p]; [-p,0] ]

and

Q = [

[0,q]; [-q,0] ]

belong to A, and t is a real number, then

P+Q, and tP

belong to A as well.

Notice that

P+Q = [ [0,p+q]; [-p-q,0] ] = [

[0,p+q]; [-(p+q),0] ]

and

tP = [ [0,tp]; [-tp,0] ]

Both matrices

are antisymmetric, so they belong to A,

and therefore A is a subspace of M2,2

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