# Linear Alg. Subspaces

I am brand new to learning about subspaces in my linear Algebra class. Ive tried to follow khan academy but to no avail. I encountered these practice problems in the textbook. However my textbook inconveniently provides answers for odd numbered problems only.

I want to further my understanding. Could someone please explain how to go about these problems?

## Given vector space V = the set of all 2 × 2 matrices.....

*1) Show that the sum of two 2 × 2 nonsingular matrices may be
singular.

2) Show that the sum of two 2 × 2 singular matrices may be nonsingular.*


Thank you

• What did you try for these problems? Did you try to come up with an example for either? – Omnomnomnom Oct 13 '16 at 2:12
• Ive been sitting at the problems for about an hour now. I cant seem to make sense of them given the info in my text book :( – Rebecca Elwood Oct 13 '16 at 2:13
• @RebeccaElwood did you write things down? – qbert Oct 13 '16 at 2:14
• Well, you didn't really answer my question, so let me ask this: do you understand what the question is telling you to do? Do you know what a "singular matrix" is? – Omnomnomnom Oct 13 '16 at 2:16
• A square matrix that has the property of the determinant being 0? – Rebecca Elwood Oct 13 '16 at 2:37

Hint: Instead of summing two random non-singular matrices hoping to get a singular matrix, try from the other direction: Take an obviously singular matrix. For example both the rows are identical, say $\displaystyle A={2\quad 3\choose 2\quad3}$. Change a single entry, call the resulting matrix $A_1$ and define $A_2= (A-A_1)$ . Now your experimentation is focussed: what changes will make both $A_1$ and $A_2$ non-singular.
1) You want to find some $A$ and $B$ nonsingular (i.e. invertible/determinant nonzero) whose sum is not invertible. I would go with I and -I, but maybe try and find your own.
2) Similarly, you just need to find an example $$\begin{bmatrix}1&0\\0&0\end{bmatrix},\begin{bmatrix}0&0\\0&1\end{bmatrix}$$ do nicely.