I'm trying to prove that the trace of a nilpotent matrix is $0$. Here's what I have:  Let $A \in M_n(k)$, where $k$ is a field. If $A$ is nilpotent then, in fact, $A^n=0$, (this result was proven in the previous problem.) By Proposition 20 on page 428 (Dummit &amp; Foote) the characteristic polynomial of $A$ divides some power of the minimal polynomial of $A$, and so is of the form $x^m$ for some $m \ge t$. Thus, the eigenvalues of $A$ are all equal to $0$ and the only solution to the equation $xv-Av=0$ is the trivial solution. Since it is assumed that $v \neq 0$, then it must be that $A=0$.   This is a much stronger conclusion than the conclusion I'm aiming for and so I am suspicious. I probably made an error. Please help.
Does anyone know a proof from the first principles that a nilpotent matrix has zero trace. No eigenvalues, no characteristic polynomials, just definition and basic facts about bases and matrices.
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I have an issue with my Minecraft launcher - I downloaded modpack for minecraft then the launcher started an update for the modpack version which it's 1.6.2, however then I started to get the issue      [15:40:59 ERROR]: Gave up trying to download      for job 'Version &amp; Libraries'      [15:40:59 ERROR]: Job 'Version &amp; Libraries' finished with 26   failure(s)! (took 0:01:06.362)   At the same time I had just downloaded Avira antivirus and ran it, but unfortunately  removing it and trying again didn't help. I've tried to restore the system to the point before downloading the antvirus and the mod pack a 2 days ago, I've tried turning off the firewall and I have tried removing the whole .minecraft folder all with no luck at all. I can't figure it out, all the servers are working and I have good internet connection but some of the files download fine and other don't.  The full log is here:  [15:40:59 WARN]: Couldn't download https ://libraries.minecraft.net/net/sf/jopt-simple/jopt-simple/4.5/jopt-simple-4.5.jar for job 'Version &amp; Libraries' java.net.SocketTimeoutException: connect timed out         at java.net.TwoStacksPlainSocketImpl.socketConnect(Native Method) ~[?:1.7.0_45]         at java.net.AbstractPlainSocketImpl.doConnect(Unknown Source) ~[?:1.7.0_45]         at java.net.AbstractPlainSocketImpl.connectToAddress(Unknown Source) ~[?:1.7.0_45]         at java.net.AbstractPlainSocketImpl.connect(Unknown Source) ~[?:1.7.0_45]         at java.net.PlainSocketImpl.connect(Unknown Source) ~[?:1.7.0_45]         at java.net.SocksSocketImpl.connect(Unknown Source) ~[?:1.7.0_45]         at java.net.Socket.connect(Unknown Source) ~[?:1.7.0_45]         at sun.security.ssl.SSLSocketImpl.connect(Unknown Source) ~[?:1.7.0_45]         at sun.net.NetworkClient.doConnect(Unknown Source) ~[?:1.7.0_45]         at sun.net.www.http.HttpClient.openServer(Unknown Source) ~[?:1.7.0_45]         at sun.net.www.http.HttpClient.openServer(Unknown Source) ~[?:1.7.0_45]         at sun.net.www.protocol.https.HttpsClient.&lt;init&gt;(Unknown Source) ~[?:1.7.0_45]         at sun.net.www.protocol.https.HttpsClient.New(Unknown Source) ~[?:1.7.0_45]         at sun.net.www.protocol.https.AbstractDelegateHttpsURLConnection.getNewHttpClient(Unknown Source) ~[?:1.7.0_45]         at sun.net.www.protocol.http.HttpURLConnection.plainConnect(Unknown Source) ~[?:1.7.0_45]         at sun.net.www.protocol.https.AbstractDelegateHttpsURLConnection.connect(Unknown Source) ~[?:1.7.0_45]         at sun.net.www.protocol.http.HttpURLConnection.getInputStream(Unknown Source) ~[?:1.7.0_45]         at java.net.HttpURLConnection.getResponseCode(Unknown Source) ~[?:1.7.0_45]         at sun.net.www.protocol.https.HttpsURLConnectionImpl.getResponseCode(Unknown Source) ~[?:1.7.0_45]         at net.minecraft.launcher.updater.download.ChecksummedDownloadable.download(ChecksummedDownloadable.java:49) ~[launcher.jar:?]         at net.minecraft.launcher.updater.download.DownloadJob.popAndDownload(DownloadJob.java:108) [launcher.jar:?]         at net.minecraft.launcher.updater.download.DownloadJob.access$000(DownloadJob.java:12) [launcher.jar:?]         at net.minecraft.launcher.updater.download.DownloadJob$1.run(DownloadJob.java:89) [launcher.jar:?]         at java.util.concurrent.Executors$RunnableAdapter.call(Unknown Source) [?:1.7.0_45]         at java.util.concurrent.FutureTask.run(Unknown Source) [?:1.7.0_45]         at java.util.concurrent.ThreadPoolExecutor.runWorker(Unknown Source) [?:1.7.0_45]         at java.util.concurrent.ThreadPoolExecutor$Worker.run(Unknown Source) [?:1.7.0_45]         at java.lang.Thread.run(Unknown Source) [?:1.7.0_45] [15:40:59 ERROR]: Gave up trying to download https ://libraries.minecraft.net/net/sf/jopt-simple/jopt-simple/4.5/jopt-simple-4.5.jar for job 'Version &amp; Libraries' [15:40:59 ERROR]: Job 'Version &amp; Libraries' finished with 26 failure(s)! (took 0:01:06.362)