# Let $M_{2\times 2}$ be the vector space of all $2\times 2$ matrices. Show that the set of non-singular matrices is NOT a subspace.

I am working on problems from my textbook. However, I am lost as to how to show this.

A. Let $M_{2\times 2}$ be the vector space of all $2\times 2$ matrices. Show that the set of non-singular $2\times 2$ matrices is NOT a subspace of $M_{2\times 2}$.

B. Let $M_{2\times 2}$ be the vector space of all $2\times 2$ matrices. Show that the set of singular $2\times 2$ matrices is NOT a subspace of$M_{2\times 2}$.

C. Describe the smallest subspace of $M_{2\times 2}$ that contains matrices $$\begin{pmatrix}2 & 1 \\ 0 & 0\end{pmatrix},\ \ \begin{pmatrix}1 & 0 \\ 0 & 2\end{pmatrix}\ \ \text{and}\ \ \ \begin{pmatrix}0 & -1 \\ 0 & 0\end{pmatrix}$$

Find the dimension of the subspace.

I think I can prove that addition for A and B is not closed, thus disproving the potential for subspace. Though, I am not sure about C.

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