I want to prove the following question but I have no idea to start!  let $A$ be a $3\times2$ matrix and $B$ be a $2\times3$ one. if $AB=\begin{bmatrix}8 &amp; 2 &amp;-2 \\2 &amp; 5 &amp; 4 \\-2 &amp; 4 &amp; 5 \end{bmatrix}$ then prove that $BA=\begin{bmatrix}9 &amp; 0\\ 0 &amp; 9\end{bmatrix}$. [ hint:if $AB=I$ then we have $BA=I$]
Let $A∈\mathscr{M}_{3×2}(\mathbb{R})$ and $B∈\mathscr{M}_{2\times3}(\mathbb{R})$ be matrices satisfying   $AB =\begin{bmatrix} 8 &amp;2 &amp;−2\\ 2 &amp;5 &amp;4\\ −2 &amp;4&amp; 5 \end{bmatrix}$. Calculate $BA$. (Golan, The Linear Algebra a Beginning Graduate Student Ought to Know, Exercise 426.)   Maybe it can be solved by solving a system of equations, but I think there is a shorter way since this problem was in my exam.  Thanks.
Why use a ReentrantLock if one can use synchronized(this)? Sort by author in Alpha bibliography Using the alpha bibliography style, citations are sorted by the citation key. I'd like to use the same style of authors-year citation keys (e.g., [CP92]) but have the bibliography sorted by the first author's lastname instead of alphabetically by the citation key itself. This would keep the same papers by the same first author together.  For example, alpha would produce the following sorted order:        [Cha81] D. Chaum. Untraceable...   [CL04] J. Camenisch and A. Lysyanskaya. Signature...   [CP92] D. Chaum and T. Pedersen. Wallet...      Whereas, abbr would produce the following sorted order:        [1] J. Camenisch and A. Lysyanskaya. Signature...   [2] D. Chaum. Untraceable...   [3] D. Chaum and T. Pedersen. Wallet...      I am looking for the same sorting as abbr but maintaining the citation key style of alpha.  Is there a way to achieve this hybrid style? How exactly using VB6 can I can call any Windows shell command as you would from the command-line?  For example, something as trivial as:  echo foo ElasticSearch doesn't start after installing 5.6.4 I tried to use sudo cd name_of_dir but am getting the error message:   sudo: cd: command not found   Is there any other way to enter a directory owned by another user which has 700 permission? "every" vs "each"? When to use icons vs. icons with text vs. just text links? Assume a table structure of MyTable(KEY, datafield1, datafield2...).  Often I want to either update an existing record, or insert a new record if it doesn't exist.  Essentially:  IF (key exists)   run update command ELSE   run insert command   What's the best performing way to write this? Assuming file2 already exists, the command   &gt; file1 &lt; file2 cat   appears to be copying the content of file2 to file1.  But I cannot understand this structure.  I understand that "Nothing" is being directed to file1 (creating or erasing its content). Then the content of file2 is being directed to file1.  Why is cat after file2? How does it know to cat file2 if the operands are not in the correct order? Comment hint not disappearing when I start to type comment If I click "add comment", the comment hint (placeholder; "Request clarification or add information.") is not a hint, but editable text. If I delete the text and click outside of the comment then the hint appears as it should and when I click back into the comment box the hint disappears as it should.   I am on Internet Explorer 10.0.7.  Am I just oblivious and this is how it has always been (having to delete the comment hint), or is this a bug? In PHP you can just use $_POST for POST and $_GET for GET (Query string) variables. What's the equivalent in Python? $4$ letter words taken from the letters of CONCENTRATIONS How many words with or without meaning can be made from the letters of the word CONCENTRATIONS by taking $4$ letters at a time?  There are  $14$ letters total present in the word. $4$ letters can be picked in $14\choose 4$ ways. $4$ letters can be arranged in $4!$ ways. So ideally answer has to be ${14\choose 4} * 4!$. But I know this answer is wrong because some letters are repeating. How do I solve this then? I appreciate any help. For what powers $k$ is the polynomial $n^k-1$ divisible by $(n-1)^2$? I dont know how to sort a list of SObject Case.  I have a List&lt;Case&gt; newCases which i iterate over in for cycle, where i save Cases to my new List&lt;Case&gt; casesList.  for (Case newCase : newCases) {             if(newCase.Case_Content__c == '111' &amp;&amp; newCase.Status == '24'){                 String cuid = newCase.CUID__c;                 cuidSet.add(cuid);                 casesList.add(newCase);                 casesList.sort(); //TODO sorting by CUID__c             }         }   What i need to do is, to sort this new list by field CUID__c. So the Cases with same CUID are together.  Thank you for any help :) Should I use exit() or just return statements in main()? Personally I favor the return statements because I feel it's like reading any other function and the flow control when I'm reading the code is smooth (in my opinion). And even if I want to refactor the main() function, having return seems like a better choice than exit().  Does exit() do anything special that return doesn't? \sqrt[x]{y} Breaks With unicode-math After running Win 7 for some unknown period of time, the taskbar icon highlight starts to stick (video demo here: ). If I restart or kill explorer.exe and relaunch, it goes away.  Any idea what might be causing this? Are there systems in classical mechanics that are provably non-deterministic? If $f \in L_p \cap L_q$, $1 \leq p < r < q < \infty$ then $f \in L_r$. If $f \in L_p \cap L_q$ and $1 \leq p &lt; r &lt; q &lt; \infty$ then $f \in L_r$.   Proof: Let $E = \{ x \in X: \left\lvert f \right\rvert \leq 1 \}$. Then, $$\int \left\lvert f \right\rvert^r \mathrm{d}\mu = \int_E \left\lvert f \right\rvert^r \mathrm{d}\mu+\int_{E^C} \left\lvert f \right\rvert^r \mathrm{d}\mu \leq \int_E \left\lvert f \right\rvert^p \mathrm{d}\mu+\int_{E^C} \left\lvert f \right\rvert^q \mathrm{d}\mu$$ $$\leq \int \left\lvert f \right\rvert^p \mathrm{d}\mu+\int \left\lvert f \right\rvert^q \mathrm{d}\mu &lt; \infty.$$ Thus, $f \in L_r$.  I just want to make sure I am somehow not missing anything. Thanks Javascript profanity match NOT replace