A square matrix A is idempotent if A2=A. Let V be the vector space of all 2×2 matrices with real entries. Let H be the set of all 2×2 idempotent matrices with real entries. Is H a subspace of the vector space V?

No, H is not a subspace of the vector space V.

Step-by-step explanation:

A matrix is a rectangular array in which elements are arranged in rows and columns.

A matrix in which number of columns is equal to number of rows is known as a square matrix.

Let H denote set of all 2×2  idempotent matrices.

H is a subspace of a vector space V if for and   .

Let

As , A is idempotent.

So,

So, A+A is not idempotent and hence, does not belong to H.

So, H is not a subspace of the vector space V.