Why is the set of all 2x2 singular matrices not a vector space?

[xvtsx]

Homework Statement

The set of all 2x2 singular matrices is not a vector space. why?
$$\begin{bmatrix} 1 & 0\\ 0&0 \end{bmatrix}+\begin{bmatrix} 0 & 1\\ 0& 1 \end{bmatrix}=\begin{bmatrix} 1 & 1\\ 0 & 1 \end{bmatrix}$$

Homework Equations

Is it because the determinant in both are zero, but by performing addition you get a nonsingular matrix from a two singular matrices.

The Attempt at a Solution

det(0)+det(0)=0
c*det(0) = 0

 

[arkajad]

$$\begin{pmatrix}1&0\\0&0\end{pmatrix}+\begin{pmatrix}0&0\\0&1\end{pmatrix}=\ldots$$

 

[xvtsx]

$$\begin{pmatrix}1&0\\0&0\end{pmatrix}+\begin{pmatrix}0&0\\0&1\end{pmatrix}=\ldots$$

Sorry, but can you explain what you meant? Thanks

 

[arkajad]

Can you add these two matrices? Are they both singular? Is their sum singular? Is the set of singular matrices a vector space?

 

[xvtsx]

They are both singular and if you add them up the result would be a nonsingular matrix. Singular matrices don't have a inverse, so they aren't vector spaces.

 

[arkajad]

They are both singular and if you add them up the result would be a nonsingular matrix. Singular matrices don't have a inverse, so they aren't vector spaces.

The last sentence is not a good one. In fact it is very very bad (it would be a good exercise for you to find out why it is so bad). A good one is:

In a vector space, for any two vectors from this space, their sum should be again a vector in the same space.

The examples show that this is not the case with singular matrices: one can find examples of two singular matrices whose sum is not a singular matrix. Therefore the set of all singular matrices does not satisfy one of the necessary requirements to be a vector space. Therefore it is not a vector space.