Question

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?

Answer 1

Answer:

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 $u+v \in H$ for $u,v \in V$ and   $cu \in H$.

Let $A=\begin {pmatrix}1&0\\0&1 \end{pmatrix}$

As $A^2=A\times A=\begin {pmatrix}1&0\\0&1 \end{pmatrix}\begin {pmatrix}1&0\\0&1 \end{pmatrix}=\begin {pmatrix}1&0\\0&1 \end{pmatrix}=A$, A is idempotent.

So, $A \in H$

$A+A=\begin {pmatrix}1&0\\0&1 \end{pmatrix}+\begin {pmatrix}1&0\\0&1 \end{pmatrix}=\begin {pmatrix}2&0\\0&2\end{pmatrix} \\ \left ( A+A \right )^2=\begin {pmatrix}2&0\\0&2\end{pmatrix}\begin {pmatrix}2&0\\0&2\end{pmatrix}=\begin {pmatrix}4&0\\0&4\end{pmatrix}\neq A$So, A+A is not idempotent and hence, does not belong to H.

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