# (Linear Algebra) Vector Space

1. Oct 21, 2010

### xvtsx

1. The problem statement, all variables and given/known data
The set of all 2x2 singular matrices is not a vector space. why?
$$\begin{bmatrix} 1 & 0\\ 0&0 \end{bmatrix}+\begin{bmatrix} 0 & 1\\ 0& 1 \end{bmatrix}=\begin{bmatrix} 1 & 1\\ 0 & 1 \end{bmatrix}$$

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

3. The attempt at a solution
det(0)+det(0)=0
c*det(0) = 0

2. Oct 21, 2010

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

3. Oct 21, 2010

### xvtsx

Sorry, but can you explain what you meant? Thanks

4. Oct 21, 2010

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

5. Oct 21, 2010

### 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.

6. Oct 21, 2010