User:A2569875/模板沙盒/數學測試/2

• 1,0.4406197005382-1.56961032189i,0.4406197005382+1.56961032189i,-1.8812394010764

矩陣

• ${\begin{bmatrix}{1}&{2}\\{3}&{1}\end{bmatrix}}\$, ${\begin{bmatrix}{7}&{4}\\{6}&{7}\end{bmatrix}}\$, ${\begin{bmatrix}{19}&{18}\\{27}&{19}\end{bmatrix}}\$, ${\begin{bmatrix}{73}&{56}\\{84}&{73}\end{bmatrix}}\$, ${\begin{bmatrix}{241}&{202}\\{303}&{241}\end{bmatrix}}\$, ${\begin{bmatrix}{847}&{684}\\{1026}&{847}\end{bmatrix}}\$, ${\begin{bmatrix}{2899}&{2378}\\{3567}&{2899}\end{bmatrix}}\$, ${\begin{bmatrix}{10033}&{8176}\\{12264}&{10033}\end{bmatrix}}\$, ${\begin{bmatrix}{34561}&{28242}\\{42363}&{34561}\end{bmatrix}}\$, ${\begin{bmatrix}{119287}&{97364}\\{146046}&{119287}\end{bmatrix}}\$,
• ${\begin{bmatrix}{1}\end{bmatrix}}\$, ${\begin{bmatrix}{1}&{0}\\{0}&{1}\end{bmatrix}}\$, ${\begin{bmatrix}{1}&{0}&{0}\\{0}&{1}&{0}\\{0}&{0}&{1}\end{bmatrix}}\$, ${\begin{bmatrix}{1}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}\\{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{1}\end{bmatrix}}\$, ${\begin{bmatrix}{1}&{0}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}&{0}\\{0}&{0}&{1}&{0}&{0}\\{0}&{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{0}&{1}\end{bmatrix}}\$,
• ${\begin{bmatrix}{1}&{0}\\{0}&{1}\end{bmatrix}}\$, ${\begin{bmatrix}{2}&{0}\\{0}&{2}\end{bmatrix}}\$, ${\begin{bmatrix}{3}&{0}\\{0}&{3}\end{bmatrix}}\$, ${\begin{bmatrix}{4}&{0}\\{0}&{4}\end{bmatrix}}\$, ${\begin{bmatrix}{5}&{0}\\{0}&{5}\end{bmatrix}}\$, ${\begin{bmatrix}{6}&{0}\\{0}&{6}\end{bmatrix}}\$, ${\begin{bmatrix}{7}&{0}\\{0}&{7}\end{bmatrix}}\$, ${\begin{bmatrix}{8}&{0}\\{0}&{8}\end{bmatrix}}\$, ${\begin{bmatrix}{9}&{0}\\{0}&{9}\end{bmatrix}}\$, ${\begin{bmatrix}{10}&{0}\\{0}&{10}\end{bmatrix}}\$,

• \begin{bmatrix}{3} & {6} & {9} \\

{9} & {12} & {15} \\ {15} & {18} & {21} \\ {21} & {24} & {27}\end{bmatrix}

• {{29,32,35,38},{65,72,79,86},{101,112,123,134}}
• ${{\begin{bmatrix}{1}&{2}\\{3}&{4}\\{5}&{6}\end{bmatrix}}\times {\begin{bmatrix}{7}&{8}&{9}&{10}\\{11}&{12}&{13}&{14}\end{bmatrix}}}={\begin{bmatrix}{29}&{32}&{35}&{38}\\{65}&{72}&{79}&{86}\\{101}&{112}&{123}&{134}\end{bmatrix}}$
• $rows\left({{\begin{bmatrix}{1}&{2}\\{3}&{4}\\{5}&{6}\end{bmatrix}}\times {\begin{bmatrix}{7}&{8}&{9}&{10}\\{11}&{12}&{13}&{14}\end{bmatrix}}}\right)=3$
• {{3,6},{9,12},{15,18}}
• ${{\begin{bmatrix}{1}&{2}\\{3}&{4}\\{5}&{6}\end{bmatrix}}\times {3}}={\begin{bmatrix}{3}&{6}\\{9}&{12}\\{15}&{18}\end{bmatrix}}$
• {{1,3,5},{2,4,6}}
• ${\begin{bmatrix}{1}&{2}\\{3}&{4}\\{5}&{6}\end{bmatrix}}^{\mathrm {T} }={\begin{bmatrix}{1}&{3}&{5}\\{2}&{4}&{6}\end{bmatrix}}$
• $I_{3}={\begin{bmatrix}{1}&{0}&{0}\\{0}&{1}&{0}\\{0}&{0}&{1}\end{bmatrix}}$
• $\operatorname {diag} \left(1,\,2,\,3\right)={\begin{bmatrix}{1}&{0}&{0}\\{0}&{2}&{0}\\{0}&{0}&{3}\end{bmatrix}}$
• ${{\begin{bmatrix}{1}&{2}\\{3}&{4}\\{5}&{6}\end{bmatrix}}\times {i}}={\begin{bmatrix}{i}&{2i}\\{3i}&{4i}\\{5i}&{6i}\end{bmatrix}}$

• $selectlist\left(2,\,3,\,{\begin{bmatrix}{7}&{8}&{9}\\{10}&{11}&{12}\\{13}&{14}&{15}\end{bmatrix}}\right)=12$

• ${{{\begin{bmatrix}{1}&{2}\\{3}&{4}\\{5}&{6}\end{bmatrix}}\times {\begin{bmatrix}{7}&{8}&{9}&{10}\\{11}&{12}&{13}&{14}\end{bmatrix}}}\times {4}}={\begin{bmatrix}{116}&{128}&{140}&{152}\\{260}&{288}&{316}&{344}\\{404}&{448}&{492}&{536}\end{bmatrix}}$
• $\operatorname {trunc} \left({\begin{bmatrix}{1}&{2}&{3}&{0}\\{5}&{7}&{6}&{2}\\{7}&{9}&{8}&{1}\\{2}&{1}&{3}&{9}\end{bmatrix}}^{-1},\,5\right)={\begin{bmatrix}{-0.48809}&{-1.21428}&{1.03571}&{0.15476}\\{-0.13095}&{1.35714}&{-0.89285}&{-0.20238}\\{0.58333}&{-0.5}&{0.25}&{0.08333}\\{-0.07142}&{0.28571}&{-0.21428}&{0.07142}\end{bmatrix}}$
• $\det {\begin{bmatrix}{1}&{2}&{3}&{0}\\{5}&{7}&{6}&{2}\\{7}&{9}&{8}&{1}\\{2}&{1}&{3}&{9}\end{bmatrix}}=-84$
• ${{\begin{bmatrix}{1}&{2}&{3}&{0}\\{5}&{7}&{6}&{2}\\{7}&{9}&{8}&{1}\\{2}&{1}&{3}&{9}\end{bmatrix}}\,{{\begin{bmatrix}{1}&{2}&{3}&{0}\\{5}&{7}&{6}&{2}\\{7}&{9}&{8}&{1}\\{2}&{1}&{3}&{9}\end{bmatrix}}^{-1}}}={\begin{bmatrix}{0.99999999999999}&{1.0014211682119\times 10^{-13}}&{1.9984014443253\times 10^{-14}}&{-9.9920072216264\times 10^{-16}}\\{-2.139954879965\times 10^{-14}}&{1.0000000000004}&{9.9420471855183\times 10^{-14}}&{-1.0130785099705\times 10^{-14}}\\{-3.0905833448003\times 10^{-14}}&{4.9021897652324\times 10^{-13}}&{1.0000000000001}&{-1.4946377469016\times 10^{-14}}\\{-8.7707618945387\times 10^{-15}}&{1.0969003483297\times 10^{-13}}&{6.9944050551385\times 10^{-14}}&{0.99999999999999}\end{bmatrix}}$
  • 囧rz……
    • $\operatorname {trunc} \left({{\begin{bmatrix}{1}&{2}&{3}&{0}\\{5}&{7}&{6}&{2}\\{7}&{9}&{8}&{1}\\{2}&{1}&{3}&{9}\end{bmatrix}}\,{{\begin{bmatrix}{1}&{2}&{3}&{0}\\{5}&{7}&{6}&{2}\\{7}&{9}&{8}&{1}\\{2}&{1}&{3}&{9}\end{bmatrix}}^{-1}}},\,8\right)={\begin{bmatrix}{0.99999999}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}\\{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{0.99999999}\end{bmatrix}}$
  • $\mathrm {cof} {\begin{bmatrix}{1}&{2}&{3}&{0}\\{5}&{7}&{6}&{2}\\{7}&{9}&{8}&{1}\\{2}&{1}&{3}&{9}\end{bmatrix}}={\begin{bmatrix}{41}&{11}&{-49}&{6}\\{102}&{-114}&{42}&{-24}\\{-87}&{75}&{-21}&{18}\\{-13}&{17}&{-7}&{-6}\end{bmatrix}}$
  • $\left(\mathrm {cof} {\begin{bmatrix}{1}&{2}&{3}&{0}\\{5}&{7}&{6}&{2}\\{7}&{9}&{8}&{1}\\{2}&{1}&{3}&{9}\end{bmatrix}}\right)_{{2},\,{3}}={\begin{bmatrix}{1}&{2}&{0}\\{7}&{9}&{1}\\{2}&{1}&{9}\end{bmatrix}}$
    • $\det \left(\mathrm {cof} {\begin{bmatrix}{1}&{2}&{3}&{0}\\{5}&{7}&{6}&{2}\\{7}&{9}&{8}&{1}\\{2}&{1}&{3}&{9}\end{bmatrix}}\right)_{{2},\,{3}}=-42$
    • $\operatorname {adj} {\begin{bmatrix}{1}&{2}&{3}&{0}\\{5}&{7}&{6}&{2}\\{7}&{9}&{8}&{1}\\{2}&{1}&{3}&{9}\end{bmatrix}}={\begin{bmatrix}{41}&{102}&{-87}&{-13}\\{11}&{-114}&{75}&{17}\\{-49}&{42}&{-21}&{-7}\\{6}&{-24}&{18}&{-6}\end{bmatrix}}$
  • $GaussElimination\left({\begin{bmatrix}{1}&{2}&{3}&{0}\\{5}&{7}&{6}&{2}\\{7}&{9}&{8}&{1}\\{2}&{1}&{3}&{9}\end{bmatrix}}\right)={\begin{bmatrix}{7}&{9}&{8}&{1}\\{0}&{-1.5714285714286}&{0.71428571428571}&{8.7142857142857}\\{0}&{0}&{2.1818181818182}&{3.8181818181818}\\{0}&{0}&{0}&{3.5}\end{bmatrix}}$
• ${{\begin{bmatrix}{1}&{2}\\{3}&{4}\\{5}&{6}\end{bmatrix}}\,{{\begin{bmatrix}{7}&{11}\\{8}&{12}\\{9}&{13}\\{10}&{14}\end{bmatrix}}^{\mathrm {T} }}}={\begin{bmatrix}{29}&{32}&{35}&{38}\\{65}&{72}&{79}&{86}\\{101}&{112}&{123}&{134}\end{bmatrix}}$
• ${\frac {\begin{bmatrix}{1}&{5}\\{7}&{9}\end{bmatrix}}{\begin{bmatrix}{2}&{6}\\{3}&{1}\end{bmatrix}}}={\begin{bmatrix}{0.875}&{-0.25}\\{1.25}&{1.5}\end{bmatrix}}$
• ${{\begin{bmatrix}{1}&{5}\\{7}&{9}\end{bmatrix}}^{\begin{bmatrix}{2}&{6}\\{3}&{1}\end{bmatrix}}}={\begin{bmatrix}{-364050.31197887+964690.04071155i}&{-594639.58668283+1155986.4299034i}\\{446972.21255032+511470.08777392i}&{486336.94902566+704877.87824775i}\end{bmatrix}}$

• $\log _{\begin{bmatrix}{1}&{5}\\{7}&{9}\end{bmatrix}}\left({{\begin{bmatrix}{1}&{5}\\{7}&{9}\end{bmatrix}}^{\begin{bmatrix}{2}&{6}\\{3}&{1}\end{bmatrix}}}\right)={\begin{bmatrix}{2.0735030509264+2.2795054463381i}&{4.109459527015+3.5442394256485i}\\{1.8699259345042-1.9535647184507i}&{2.5705958298843-1.6874506503923i}\end{bmatrix}}$
• $\log _{7}\left({{7}^{3}}\right)=3$

• ${\frac {\begin{bmatrix}{24}&{15}\\{17}&{30}\end{bmatrix}}{\begin{bmatrix}{1}&{5}\\{7}&{9}\end{bmatrix}}}={\begin{bmatrix}{-4.2692307692308}&{4.0384615384615}\\{2.1923076923077}&{2.1153846153846}\end{bmatrix}}$
• $\operatorname {round} \left({\frac {\begin{bmatrix}{24}&{15}\\{17}&{30}\end{bmatrix}}{\begin{bmatrix}{1}&{5}\\{7}&{9}\end{bmatrix}}},\,3\right)={\begin{bmatrix}{-4.269}&{4.038}\\{2.192}&{2.115}\end{bmatrix}}$
• ${\begin{bmatrix}{24}&{15}\\{17}&{30}\end{bmatrix}}\%{\begin{bmatrix}{1}&{5}\\{7}&{9}\end{bmatrix}}$ = {{1,4},{1,2}}
• ${{\begin{bmatrix}{1}&{5}\\{7}&{9}\end{bmatrix}}\,{{\begin{bmatrix}{1}&{5}\\{7}&{9}\end{bmatrix}}^{-1}}}={\begin{bmatrix}{1}&{0}\\{-1.9984014443253\times 10^{-14}}&{0.99999999999999}\end{bmatrix}}$
• ${\overline {\begin{bmatrix}{{3}+{i}}&{5}\\{{2}-{2i}}&{i}\end{bmatrix}}}={\begin{bmatrix}{3-i}&{2+2i}\\{5}&{-i}\end{bmatrix}}$：共轭转置

特征值和特征向量

• $eigenvalue\left({\begin{bmatrix}{1}&{2}&{3}\\{5}&{7}&{6}\\{7}&{9}&{8}\end{bmatrix}},\,1\right)=-1.1279462354857$
• $eigenvector\left({\begin{bmatrix}{1}&{2}&{3}\\{5}&{7}&{6}\\{7}&{9}&{8}\end{bmatrix}},\,1\right)={\begin{bmatrix}{-1.6973852602632}\\{0.30597228737293}\\{1}\end{bmatrix}}$
• ${{\begin{bmatrix}{1}&{2}&{3}\\{5}&{7}&{6}\\{7}&{9}&{8}\end{bmatrix}}\times {eigenvector\left({\begin{bmatrix}{1}&{2}&{3}\\{5}&{7}&{6}\\{7}&{9}&{8}\end{bmatrix}},\,1\right)}}={\begin{bmatrix}{1.9145593144827}\\{-0.34512028970549}\\{-1.127946235486}\end{bmatrix}}$
• ${{eigenvector\left({\begin{bmatrix}{1}&{2}&{3}\\{5}&{7}&{6}\\{7}&{9}&{8}\end{bmatrix}},\,1\right)}\times {eigenvalue\left({\begin{bmatrix}{1}&{2}&{3}\\{5}&{7}&{6}\\{7}&{9}&{8}\end{bmatrix}},\,1\right)}}={\begin{bmatrix}{1.9145593144828}\\{-0.34512028970524}\\{-1.1279462354857}\end{bmatrix}}$
• $\operatorname {round} \left({{{eigenvector\left({\begin{bmatrix}{1}&{2}&{3}\\{5}&{7}&{6}\\{7}&{9}&{8}\end{bmatrix}},\,1\right)}\times {eigenvalue\left({\begin{bmatrix}{1}&{2}&{3}\\{5}&{7}&{6}\\{7}&{9}&{8}\end{bmatrix}},\,1\right)}}-{{\begin{bmatrix}{1}&{2}&{3}\\{5}&{7}&{6}\\{7}&{9}&{8}\end{bmatrix}}\times {eigenvector\left({\begin{bmatrix}{1}&{2}&{3}\\{5}&{7}&{6}\\{7}&{9}&{8}\end{bmatrix}},\,1\right)}}},\,10\right)={\begin{bmatrix}{0}\\{0}\\{0}\end{bmatrix}}$

矩陣函數

• ${\begin{bmatrix}{1}&{2}&{3}\\{5}&{7}&{6}\\{7}&{9}&{8}\end{bmatrix}}^{-1}={\begin{bmatrix}{-0.33333333333333}&{-1.8333333333333}&{1.5}\\{-0.33333333333333}&{2.1666666666667}&{-1.5}\\{0.66666666666667}&{-0.83333333333333}&{0.5}\end{bmatrix}}$
• $\operatorname {round} \left({\sqrt {\begin{bmatrix}{1}&{2}&{3}\\{5}&{7}&{6}\\{7}&{9}&{8}\end{bmatrix}}},\,3\right)={\begin{bmatrix}{0.512+0.828i}&{0.344+0.358i}&{0.764-0.508i}\\{1.108-0.149i}&{2.011-0.065i}&{1.265+0.091i}\\{1.617-0.488i}&{1.98-0.211i}&{2.139+0.299i}\end{bmatrix}}$
• $\operatorname {round} \left({{\sqrt {\begin{bmatrix}{1}&{2}&{3}\\{5}&{7}&{6}\\{7}&{9}&{8}\end{bmatrix}}}\times {\sqrt {\begin{bmatrix}{1}&{2}&{3}\\{5}&{7}&{6}\\{7}&{9}&{8}\end{bmatrix}}}},\,3\right)={\begin{bmatrix}{1}&{2}&{3}\\{5}&{7}&{6}\\{7}&{9}&{8}\end{bmatrix}}$
• $\operatorname {round} \left({\sqrt {\begin{bmatrix}{1}&{2}&{3}&{5}\\{5}&{7}&{6}&{2}\\{7}&{9}&{8}&{1}\\{2}&{1}&{3}&{9}\end{bmatrix}}},\,3\right)={\begin{bmatrix}{0.44+0.97i}&{0.317+0.009i}&{0.639-0.172i}&{1.36-0.463i}\\{1.097-0.174i}&{2.039-0.002i}&{1.243+0.031i}&{0.183+0.083i}\\{1.635-0.571i}&{2.011-0.005i}&{2.169+0.101i}&{-0.254+0.272i}\\{0.335-0.005i}&{-0.045}&{0.555+0.001i}&{2.949+0.003i}\end{bmatrix}}$
• $\operatorname {round} \left({{\sqrt {\begin{bmatrix}{1}&{2}&{3}&{5}\\{5}&{7}&{6}&{2}\\{7}&{9}&{8}&{1}\\{2}&{1}&{3}&{9}\end{bmatrix}}}\times {\sqrt {\begin{bmatrix}{1}&{2}&{3}&{5}\\{5}&{7}&{6}&{2}\\{7}&{9}&{8}&{1}\\{2}&{1}&{3}&{9}\end{bmatrix}}}},\,3\right)={\begin{bmatrix}{1}&{2.061-0.021i}&{3}&{5}\\{5}&{7.008+0.004i}&{6}&{2}\\{7}&{8.989+0.012i}&{8}&{1}\\{2.049-0.008i}&{1.223}&{3.056+0.001i}&{9.008+0.004i}\end{bmatrix}}$
• ${{\begin{bmatrix}{1}&{2}&{3}&{5}\\{5}&{7}&{6}&{2}\\{7}&{9}&{8}&{1}\\{2}&{1}&{3}&{9}\end{bmatrix}}-{{\sqrt {\begin{bmatrix}{1}&{2}&{3}&{5}\\{5}&{7}&{6}&{2}\\{7}&{9}&{8}&{1}\\{2}&{1}&{3}&{9}\end{bmatrix}}}\times {\sqrt {\begin{bmatrix}{1}&{2}&{3}&{5}\\{5}&{7}&{6}&{2}\\{7}&{9}&{8}&{1}\\{2}&{1}&{3}&{9}\end{bmatrix}}}}}={\begin{bmatrix}{1}&{-0.0609069612035+0.02065600290136i}&{3}&{5}\\{5}&{-0.0081857435936996-0.0037187820564415i}&{6}&{2}\\{7}&{0.011375233491499-0.012178315529462i}&{8}&{1}\\{-0.0491448383185+0.0077259881075695i}&{-0.2233735344013-0.0002865687318072i}&{-0.0556547457662-0.0014408810380774i}&{-0.0081857435939003-0.0037187820564442i}\end{bmatrix}}$
• ${{\operatorname {round} \left({{\sqrt {\begin{bmatrix}{1}&{2}&{3}&{5}\\{5}&{7}&{6}&{2}\\{7}&{9}&{8}&{1}\\{2}&{1}&{3}&{9}\end{bmatrix}}}\times {\sqrt {\begin{bmatrix}{1}&{2}&{3}&{5}\\{5}&{7}&{6}&{2}\\{7}&{9}&{8}&{1}\\{2}&{1}&{3}&{9}\end{bmatrix}}}},\,10\right)}-{\begin{bmatrix}{1}&{2}&{3}&{5}\\{5}&{7}&{6}&{2}\\{7}&{9}&{8}&{1}\\{2}&{1}&{3}&{9}\end{bmatrix}}}={\begin{bmatrix}{0}&{0.0609069612-0.0206560029i}&{0}&{0}\\{0}&{0.0081857436000004+0.0037187821i}&{0}&{0}\\{0}&{-0.011375233500001+0.0121783155i}&{0}&{0}\\{0.0491448383-0.0077259881i}&{0.2233735344+0.0002865687i}&{0.0556547458+0.001440881i}&{0.0081857436000004+0.0037187821i}\end{bmatrix}}$
• $\operatorname {round} \left(\log _{2}{\begin{bmatrix}{1}&{2}&{3}\\{5}&{7}&{6}\\{7}&{9}&{8}\end{bmatrix}},\,3\right)={\begin{bmatrix}{0.398+3.532i}&{1.483+1.53i}&{-0.074-2.166i}\\{1.409-0.637i}&{0.52-0.276i}&{2.286+0.39i}\\{1.36-2.081i}&{2.663-0.901i}&{1.667+1.276i}\end{bmatrix}}$
• $\operatorname {round} \left({{2}^{\log _{2}{\begin{bmatrix}{1}&{2}&{3}\\{5}&{7}&{6}\\{7}&{9}&{8}\end{bmatrix}}}},\,3\right)={\begin{bmatrix}{1}&{2}&{3}\\{5}&{7}&{6}\\{7}&{9}&{8}\end{bmatrix}}$

錯誤測試

• {{0.33823557857999+2.460010873455i,0.81979946670129+1.0142959793242i,0.12214321459583-1.4565156834862i,-0.14436663080365-0.056479668696189i},{0.87893695509153-0.43650773622565i,0.64019128979017-0.17997808325771i,1.327499855358+0.25844615998903i,0.38695747530007+0.010021830631473i},{0.96913088796228-1.4546912509079i,1.7566025594487-0.59978901023376i,1.2334688116638+0.86128912861335i,-0.055701949676983+0.033398421443206i},{0.24389271791696-0.01179203978823i,-0.27314130324697-0.0048620185684112i,0.47740097115575+0.0069817947055361i,2.2189211188093+0.00027073477913376i}}
• {{0.69314718055995,0,0,0},{0,0.69314718055995,0,0},{0,0,0.69314718055995,0},{0,0,0,0.69314718055995}}
• {{0.69314718055995,0},{0,0.69314718055995}}