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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

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    $\begingroup$ What did you try for these problems? Did you try to come up with an example for either? $\endgroup$ – Omnomnomnom Oct 13 '16 at 2:12
  • $\begingroup$ 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 :( $\endgroup$ – Rebecca Elwood Oct 13 '16 at 2:13
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    $\begingroup$ @RebeccaElwood did you write things down? $\endgroup$ – qbert Oct 13 '16 at 2:14
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    $\begingroup$ 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? $\endgroup$ – Omnomnomnom Oct 13 '16 at 2:16
  • $\begingroup$ A square matrix that has the property of the determinant being 0? $\endgroup$ – Rebecca Elwood Oct 13 '16 at 2:37
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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.

Do the same for part 2.

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You just need to come up with an example for each case.

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.

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