If $A$ is a square matrix of order 3 with $\det(A)=1$ and $AA^\top=I$, prove that $\det(A-I)=0$. If $A$ is a square matrix of order 3 with $\det(A)=1$ and $AA^\top=I$, prove that $\det(A-I)=0$.   I tried everything I know but couldn't get to the proof.
Prove that $\det(M-I)=0$ if $\det(M)=1$ and $MM^T=I$ $M$ is a $3 \times 3$ matrix such that $\det(M)=1$ and $MM^T=I$, where $I$ is the identity matrix. Prove that $\det(M-I)=0$  I tried to take $M$ $=$ $$ \begin{pmatrix}         a  &amp;b &amp; c \\         d &amp; e &amp; f \\         g &amp; h &amp; i         \end{pmatrix} $$   but its expansion is too long.
If humans were able to catch all sun energy reaching the earth for their use, will the climate change? I guess that energy will be used up and, at the end, will contribute to heat the earth, so I see no big differences...  please explain your point of view. Includegraphics maximum width With \includegraphics I can give the width of the resulting image (which would be scaled).   How can I give a maximum width of the image? That is, if the original is smaller than X, I'd like to keep the smaller image, if the image is larger than X, then the image width should be X.  Using pdflatex/graphicx. Trouble plotting arctan with pgfplots I'm trying to plot arctan(x) with pgfplots but I don't get a good output. Here is my code:   \begin{cfigur}\begin{axis}[      width=160pt,compat=1.5.1,grid style={ultra thin},every axis plot post/.append style={thick},      x tick label style={font=\tiny},y tick label style={font=\tiny},      scale only axis,grid=major,axis lines=middle,      xlabel={$x$},      ylabel={$y$},      xmin=-200,      xmax=200,      domain=-200:210,      ymin=-5.5,      ymax=5.5,      xtick={-150,-100,...,150},      ytick={-5, -4,...,5},      restrict y to domain=-20:20,      legend style={at={(0.5,-0.05)},anchor=north,nodes={right}},  ]  \addplot[mark=none,color=blue, samples=500]{atan(x)};  \addlegendentry{$y = \tan^{-1}x $};  \end{axis}\end{cfigur} I've lost my portal home, and I am invincible. How can I leave the nether? Will metering from a DSLR apply to a film SLR? I still like to shoot with film (I use a Nikon F3) but I like the immediate feedback of digital.  I'm thinking of buying a used DSLR to help me get the framing and lighting right when I shoot static subjects.  Then I'd take the final shot with my F3. But would the metering be transferable?  If the shot looks perfect at 1/125 with f/2 on a DSLR, will the same settings work on a film camera? If photons have no mass, how can they have momentum? Why is my ethernet interface called enp0s10 instead of eth0? Can a Kazakh national transit in Dubai without a visa? Dynamic Paint not showing in rendered image I am a non-EU national, permanent UK Resident. Recently, my work issued me with a multiple Schengen visa with Norway as the first (main) point of entry. Now I want to go on holidays to Greece - do I need to apply to Schengen visa again, as Greece is not Norway and I am not going there for work? how to change a pose lib using script I want to make a button that changes the active poselib to another one from the list, a specific one. I look in the console and can't see any operator. any ideas? I have been using this command successfully, which changes a variable in a config file and then executes a Python script within a loop: for((i=114;i&lt;=255;i+=1)); do echo $i &gt; numbers.txt; python DoMyScript.py; done  As each DoMyScript.py instance takes about 30 seconds to run before terminating, I'd like to relegate them to the background while the next one can be spawned. I have tried what I am familiar with, by adding in an ampersand as below: for((i=114;i&lt;=255;i+=1)); do echo $i &gt; numbers.txt; python DoMyScript.py &amp;; done  However, this results in the below error: -bash: syntax error near unexpected token `;' Inverse of function is continuous or not? How to prove $(f \circ\ g) ^{-1} = g^{-1} \circ\ f^{-1}$? (inverse of composition) I'm doing exercise on discrete mathematics and I'm stuck with question:     If $f:Y\to Z$ is an invertible function, and $g:X\to Y$ is an invertible function, then the inverse of the composition $(f \circ\ g)$ is given by $(f \circ\ g) ^{-1} = g^{-1} \circ\ f^{-1}$.   I've no idea how to prove this, please help me by give me some reference or hint to its solution. ModernCV Remove horizontal line in sections I want to write my cv with moderncv-Classic style. Can you tell me, how I can remove the horizontal line in the left of each section? Can a self-complementary graph on 100 vertices have precisely one vertex of degree 50? Fancy chapter headings How do I reset my manor so I can rebuild it with different options? How can I achieve the correct garden tap orientation when threading it in? I'm installing an external tap using a wall plate like this one:    And a tap like this one:    This might be a stupid question, but if I screw the tap into the wall plate it will end up at a certain angle. I can install the wall plate with that angle and the tap will end up with the correct orientation (i.e. pointing down). But what if I want to put another tap in later that has its thread cut slightly differently? How do I prevent it ending up upside-down or whatever? Are all the threads cut exactly the same in these taps?  This seems like a stupid design. Is there any other way to do a clean install withot formatting /home? I think my Ubuntu is a bit broken from messing around with it.  Unfortunately during installation, I did not make a separate partition for /home. Now I need to re-install but want to keep my music and films.  Any ideas...