Let $A$ be a $2 \times 3$ matrix with real entries and let $B$ be a $3 \times 2$ matrix with real entries. If $\det(AB)=4$ then find the value of $\det(BA)$.  My attempt: I am aware that $\det(AB)=\det(BA)$ when $A$ and $B$ are of same order. But how to do this?
Let 'A' be a $2 \times 3$ matrix where as B be a $3 \times 2$ matrix if $\det(AB) = 4$ the find value of the $\det(BA)$  My attempt: I took A =  $$         \begin{bmatrix}         2 &amp; 0  &amp;0\\         0 &amp; 0  &amp;2\\         \end{bmatrix} $$  B= $$         \begin{bmatrix}         1 &amp; 0  \\         0 &amp; 0  \\         0 &amp; 1  \\         \end{bmatrix} $$  It satisfies given condition and I get $\det(BA)=0$   But I have not proved it  How do I prove that it is always zero (background)I am 12th grader and I know about adjoint,inverse,determinant,rank of a matrix and the other basics.  However I do NOT know about eigenvalues and eigenvectors.
If all motion is relative, how does light have a finite speed? Facebook must have changed something again because I cannot find out how to view the list of subscribers.  In Googling for this, I came upon this question:    However, it doesn't take me to my list os subscribers.  Where do I view this list now?  Please include a screenshot and highlight the link I need to click on.  This way, when Facebook changes their layout again, your screenshot will at least clue future inquirers in on this fact so they know they're not going crazy.  Edit  Here's what I see when I log into Facebook: How to mount PCIe SSD from MacBook Air in another computer I got this SSD out of an Apple laptop that's broken from liquid damage.  What is this connection called and how do I mount this storage device to retrieve data from it? It looks like PCIe. Does Apple sell a part to mount this via USB adapter or something? Why is $a^n - b^n$ divisible by $a-b$? A more specific question about a tricky redirect case Blank lines in an equation and some other similar function Why blank lines between equation makes error? there are a lot of commands in latex and almost for all of them these blank lines are not important,  but why in this case is important and is there any similar function? Factoring Quadratics Is there a method to find which numbers to use when simplifying quadratics?   For example $x^2 + 5x + 6$ is easy enough to find, but what if I have bigger numbers, or I have this quadratic expression: $x^2 + 7x – 6$, then it gets very hard to think of a valid simplification.  EDIT: Here's an example: $(x^2 + 5x + 6)/(x + 2)$. When simplifying the first half we get $(x + 2)(x + 3)/(x + 2)$, then we can cancel out $(x + 2)$ and the answer would be $(x + 3)$. The question is: How to always find the right numbers which to put in? (in this example they where 2 and 3) How can we communicate short messages long-distance? In our 5e game, we stole a boat. And a horse carriage. And a lot of other stuff for moving things between cities.   Our DM meant for this to be a way to skip road encounters. We decided to fund a shipping company and hire managers (Knights retainers, etc.) to turn a profit. The DM ruled that these would cover lifestyle expenses, provide us with easy smuggling, etc.   However, we want to keep tabs on the treasure we're moving back and forth.   What can we use for communicating with our managers from anywhere in the world? What about from say, across the city?  We don't need to communicate often or for long - a single round will work. All we need is for the manager to say "we need you back" How do you get the caret position in a &lt;textarea&gt; using JavaScript?  For example: This is| a text  This should return 7.  How would you get it to return the strings surrounding the cursor / selection?  E.g.: 'This is', '', ' a text'.  If the word “is” is highlighted, then it would return 'This ', 'is', ' a text'. Why isn't it possible to combine vendor-specific pseudo-elements/classes into one rule set? PSTricks error "Undefined control sequence  \c@lor@to@ps" There is a remote Windows server on a private network which I can connect to via Remote Desktop Connection.  I would like to be able to make TCP/IP connections from my computer to other computers on that server's network.  Remote Desktop Connection makes it possible to share printers, drives, and other local resources through the connection.  Is there any way to "tunnel" a TCP/IP connection via RDC?  I'd like something similar to the port-forwarding provided by SSH.  I don't see any way to do this via RDC, but I'm hoping the capability is there and I just don't know about it. How to specifically use ntfsundelete to recover directory This is a followup on my original post  which may be deleted in the future.  In the process of backing up my audio books, I "permanently" removed a folder (with subfolders) that I had already exclusively moved to that drive. It's path is /media/gratis/Seagate Expansion Drive/Audiobooks/Jason Dark. The sole folder in that should be called John Sinclair; but I'm not sure about that.  Which tool offers the best chances to recover those files? I haven't written plenty on that drive afterwards; so I hope for the best. It's not that other methods failed; I just cant get my head around what to do and I want to be cautious. It's a NTFS Drive.  According to testdisk it has an "EFI GPT" Partition table.  Disk /dev/sdb - 5000 GB / 4657 GiB - CHS 608001 255 63, sector size=512 - Seagate Expansion Desk, FW:9401   Information According to "Disks" (Ubuntu Standard Software):  NTFS; GUID Partition Table; [...] Partition Type: Microsoft Reserved   Attempt 1: First off, ntfsprogs seems to no longer be in repos (those mentioned by another user); but it is already installed on my machine so no problems there. I always get the error:     Access is denied because the NTFS volume is already exclusively opened. The volume may be already mounted, or another software may use it which could be identified for example by the help of the 'fuser' command. You can use force option to avoid this check, but this is not recommended and may lead to data corruption.   Although I only just plugged it in and even quit Nautilus by force (nautilus -q).   Therefore, I ran sudo ntfsundelete /dev/sdb2 -f. That gave me a list of a bunch of files, 205013 to be exact. Now, what I want is the content of the directory .../Audiobooks/Jason\ Dark/ to be restored. What do I have to do? It's contents are .m4b &amp; .opus files and folders, of course. Analogue of the Cantor-Bernstein-Schroeder theorem for general algebraic structures The Cantor-Bernstein-Schroeder theorem states that if there are two sets $A$ and $B$ such that there exist injective (alternatively, surjective, assuming choice I think) maps $A \to B$ and $B \to A$, then $A$ and $B$ are in bijection.  It then seems natural to try to strengthen the result to other structures.  If $A$ and $B$ are, say, groups (or rings, modules, etc) such that there are injective homomorphisms from each to the other, then $A$ and $B$ are isomorphic.  I'm wondering if there are any results on this, or if there are known counterexamples.  The theorem for sets alone is nontrivial, so I feel like any results about how it holds for other structures would be quite interesting. Find all entire functions $f(z)$ such that $|f(z)|=1$ for $|z|=1$  Hint: First show that $f(z)$ is a polynomial.  Clearly one can not use Cauchy Estimates to prove that $f(z)$ is a polynomial, the other way is to prove that $f(z)$ has a pole at infinity which I am not sure how to prove that, are there any other ways to prove that a given entire function is a polynomial ?  Thank you ! Fancyhdr does not apply same header/footer on chapter and non-chapter pages Adventurer enters an other-dimensional realm where he repeatedly dies unexpectedly and wakes up in reality How to get permanent token? I really need to get permanent token or dynamic generator. We set job in the server and this service will come to salesforce and get the data. To provide this, I created connected app, got the token from postman by myself and gave it. After a while (approximately 1 day), we faced with session id problem.   What can I do? Why are neural networks becoming deeper, but not wider? In recent years, convolutional neural networks (or perhaps deep neural networks in general) have become deeper and deeper, with state-of-the-art networks going from 7 layers () to 1000 layers ( in the space of 4 years. The reason behind the boost in performance from a deeper network, is that a more complex, non-linear function can be learned. Given sufficient training data, this enables the networks to more easily discriminate between different classes.  However, the trend seems to not have followed with the number of parameters in each layer. For example, the number of feature maps in the convolutional layers, or the number of nodes in the fully-connected layers, has remained roughly the same and is still relatively small in magnitude, despite the large increase in the number of layers. From my intuition though, it would seem that increasing the number of parameters per layer would give each layer a richer source of data from which to learn its non-linear function; but this idea seems to have been overlooked in favour of simply adding more layers, each with a small number of parameters.  So whilst networks have become "deeper", they have not become "wider". Why is this? What do we lose passing from the reals to the complex numbers? As normed division algebras, when we go from the complex numbers to the quaternions, we lose commutativity. Moving on to the octonions, we lose associativity. Is there some analogous property that we lose moving from the reals to the complex numbers?