用Matlab學(xué)習(xí)線(xiàn)性代數(shù)

1、用Matlab學(xué)習(xí)線(xiàn)性代數(shù)線(xiàn)性方程組與矩陣代數(shù)實(shí)驗(yàn)?zāi)康模菏煜ぞ€(xiàn)性方程組的解法和矩陣的基本運(yùn)算及性質(zhì)驗(yàn)證。Matlab 命令：本練習(xí)中用到的 Matlab 命令有：inv，floor，rand，tic，toc，rref，abs， max，round，sum， eye，triu，ones，zeroso本練習(xí)引入的運(yùn)算有：+，-，*，。其中+和-表示通常標(biāo)量及矩陣的加 法和減法運(yùn)算；*表示標(biāo)量或矩陣的乘法；對(duì)所有元素為實(shí)數(shù)的矩陣，運(yùn)算對(duì)應(yīng) 于轉(zhuǎn)置運(yùn)算。若A為一個(gè)nxn 非奇異矩陣(det!=0)且B為一個(gè)nxr矩陣，則運(yùn) 算A B等價(jià)于A-1B。實(shí)驗(yàn)內(nèi)容：用Matlab隨機(jī)生成4x4的矩陣A和B。求

2、下列指定的C,D,G,H，并確定 那些矩陣是相等的。你可以利用Matlab計(jì)算兩個(gè)矩陣的差來(lái)測(cè)試兩個(gè)矩陣 是否相等。C=A*B,D=B*A,G=(A *B)，H=(B *A) C=H;D=G;C=A *B，D=(A*B)，G=B *A，H=(B*A) C=H;D=G;C=inv(A*B),D=inv(A)*inv(B),G=inv(B*A),H=inv(B)*inv(A)C=inv(A*B) ),D=inv(A *B ),G=inv(A )*inv(B ),H=(inv(A)*inv(B) (4)中無(wú)相等的令n=200，并使用命令A(yù)=floor(10*rand(n);b=sum(A)z=one

3、s(n,1);注釋?zhuān)?n行一列全為1的矩陣)生成一個(gè)nxn矩陣和兩個(gè)Rn中的向量，它們的元素均為整數(shù)。(因?yàn)榫仃嚭?向量都很大，我們添加分號(hào)來(lái)控制輸出。方程組Ax = b的真解應(yīng)為z。為什么？【A中的每一行的元素之和 正好等于對(duì)應(yīng)b的每一列，故z為其一解，又det不等于0，RA=RAb=n，故z為其解】試說(shuō)明，可在Matlab中利用” ”運(yùn)算 或計(jì)算A-1，然后用計(jì)算A-ib來(lái)求解。比較這兩種計(jì)算方法的速度和 精度。我們將使用Matlab命令tic和toc來(lái)測(cè)量每一個(gè)計(jì)算過(guò)程消耗 的時(shí)間。只需要用下面的命令：tic,x=Ab； toctic,y=inv(A)*b; toc哪一種方法更快？tic

4、,x=Ab ；更快！為了比較這兩種方法的精度，可以測(cè)量求得的解x和y與真解z接近 的程度。利用下面的命令：max(abs(x-z)max(abs(y-z)哪種方法的到的解更精確？ max(abs(x-z)= 4.0168e-013更精確！ max(abs(y-z) = 6.1107e-013用 n=500 和 n=1000 替換(1)中的n。如(1)結(jié)果一樣!令A(yù)=floor(10*rand(6)。根據(jù)構(gòu)造，矩陣A將有整數(shù)元。將矩陣A的第六 列更改，使得矩陣A為奇異的。令B=A，A(:,6)=-sum(B(1:5,:) 設(shè)x=ones(6,1),并利用Matlab計(jì)算Ax。為什么我們知道A必為

5、奇異的？【因化簡(jiǎn)列，列成比例】試說(shuō)明。通過(guò)化為行最簡(jiǎn)形來(lái)判斷A是奇異的。令B=x*1:6，乘積AB應(yīng)為零矩陣。為什么？【因A的每一行的前 五個(gè)元素之和等于第六個(gè)元素的相反數(shù)，且在A上的每一行的元素同 乘以相同的數(shù)，則仍等于0】試說(shuō)明。用Matlab的*運(yùn)算計(jì)算AB進(jìn) 行驗(yàn)證。 令 C=floor(10*rand(6)和 D=B+C,盡管C D，但乘積 AC 和 AD 是 相等的。為什么？試說(shuō)明。計(jì)算A*C和A*D,并驗(yàn)證它們確實(shí)相等?！敬颂?B 為令 B=x*1:6; A 為 A(:,6)=-sum(B(1:5,:)】由于 A*B=0;故 AC=AD; A(B+C)=AB+AC;采用如下方式構(gòu)

6、造一個(gè)矩陣。令B=eye(10)-triu(ones(10),1),參見(jiàn)最后附表二:為什么我們知道B必為非奇異的？【上三角矩陣的行列式的值等于對(duì)角線(xiàn)上的元素相乘】令 C=inv(B)且 x=C(:,10),現(xiàn)在用B(10,1)=-1/256將B進(jìn)行微小改變。利用Matlab計(jì)算乘積Bx。由這個(gè)計(jì)算結(jié)果，你可以得出關(guān)于新矩陣B的什么結(jié)論？【化簡(jiǎn)此時(shí)B,得行最簡(jiǎn)式，RB=910，可以得出B的第10列(從19行)與x互為相反 數(shù)，且都是2的指數(shù)冪數(shù)，且第十行為0，】它是否為奇異的？【是】試說(shuō)明。用Matlab計(jì)算它的行最簡(jiǎn)形。生成一個(gè)矩陣A：A=floor(20*rand(6)并生成一個(gè)向量b：B=

7、floor(20*rand(6,1)-10因?yàn)锳是隨機(jī)生成的，我們可以認(rèn)為它是非奇異的。那么方程組Ax = b應(yīng)有唯一解。用運(yùn)算“”求解。用Matlab計(jì)算A b的行 最簡(jiǎn)形U。比較U的最后一列和解x,結(jié)果是什么？【相等】在 精確算術(shù)運(yùn)算時(shí)，它們應(yīng)當(dāng)是相等的。為什么？【行最簡(jiǎn)式中可寫(xiě) 出對(duì)應(yīng)元素的實(shí)際含義，對(duì)應(yīng)處的未知元就等于最后的數(shù)】試說(shuō)明。 為比較他們兩個(gè)，計(jì)算差U(:,7)-x或用format long考慮它們?，F(xiàn)在改變A,試它成為奇異的。令A(yù)(:,3)=A(:,1:2)*4 3【第一 列乘以4加上第二列乘以3替換到第三列上】，利用Matlab計(jì)算 rref(A b)。方程組Ax = b

8、有多少組解？【無(wú)解】試說(shuō)明。【RARAB】令y=floor(20*rand(6,1)-10且c=A*y,為什么我們知道方程組 Ax=c必為相容？的？【x此時(shí)必有一解y,故為相容的】試說(shuō)明。 計(jì)算A c的行最簡(jiǎn)形U。方程組Ax = b有多少組解？【無(wú)窮多解】 試說(shuō)明?！綬A=RA c6】由行最簡(jiǎn)形確定的自由變量應(yīng)為x3。通過(guò)考察矩陣U對(duì)應(yīng)的方程 組，可以求得x3 =0時(shí)所對(duì)應(yīng)的解。將這個(gè)解作為列向量w輸入 Matlab中。為檢驗(yàn)Aw = c，計(jì)算剩余向量c - Aw。 令U(:,7) = zeros(6,1)。矩陣U應(yīng)對(duì)應(yīng)于Ia 10】的行最簡(jiǎn)形。用U求 自變量x3 = 1時(shí)齊次線(xiàn)性方程組的解(

9、手工計(jì)算)，并將你的結(jié)果 輸入為向量Z。用A*Z檢驗(yàn)?zāi)愕慕Y(jié)論。令v = w + 3* z。向量v應(yīng)為方程組Ax = c的解。為什么？試說(shuō)明。用Matlab計(jì)算剩余向量來(lái)驗(yàn)證v為方程組的解。在這個(gè)解中，自 由變量X3的取值是什么？【x3 =3如何使用向量w和z來(lái)求所有可能的方程組的解？【v=w+n*z,其中n為任意實(shí)數(shù)】試說(shuō) 明?？紤]下圖：確定圖的鄰接矩陣A,將其輸入Matlab;計(jì)算A2并確定長(zhǎng)度為2的路的條數(shù)【72】，其起止點(diǎn)分別為：AA2+A 中的數(shù)值之和，數(shù)字表示有幾種路徑，具體看程序】 計(jì)算A4、A6、As并回答(2)中各種情況長(zhǎng)度為4、【368】6、2362 8、15800的路的條數(shù)

10、。試推測(cè)什么時(shí)候從頂點(diǎn)Vi到Vj沒(méi)有長(zhǎng)度 為偶數(shù)【即為0】的路?！緄=1, j=6; i=2,j=5; i=3,j=6或8;i=4,j=7;i=5,j=8;i=6,j=1 或 3;i=7,j=4;i=8,j=3 或 6; 計(jì)算A3、A5、A7并回答(2)中各情況長(zhǎng)度為3、154 5、9227 6098的路的條數(shù)。你由(3)得到的推測(cè)對(duì)長(zhǎng)度為奇數(shù)的路是否成立？【不成立】，試說(shuō)明【見(jiàn)程序】。推測(cè)根據(jù)i+j+k的奇偶性，是否存在長(zhǎng)度為k的路?！救鬷+j+k為偶數(shù)，不存在；相反，則存在】 【路徑見(jiàn)程序】如果我們?cè)趫D中增加邊V3,V6，V5,V8，新圖的鄰接矩陣B可首先 令 B=A，然后令 B(3,6)

11、 = 1, B(6,3) = 1, B(5,8)=1, B(8,5)=1，對(duì) k=2,3,4,5計(jì)算Bk。(4)中的推測(cè)在新的圖形中是否還是成立的？【不 成立】見(jiàn)程序】在圖中增加V6，VJ，并構(gòu)造得到的圖的鄰接矩陣C，計(jì)算C的冪次， 并驗(yàn)證你在(4)中的推測(cè)對(duì)這個(gè)新圖是否仍然成立?！静怀闪ⅰ俊疽?jiàn)程序】7 .令A(yù)=magic(8),然后計(jì)算其行最簡(jiǎn)形。使得首1對(duì)應(yīng)于前三個(gè)變量x ,x ,x ,123且其余的五個(gè)變量均為自由的。(1)令c=1:8，通過(guò)計(jì)算矩陣Ac的行最簡(jiǎn)形確定方程組Ax=c是否相容。方程組是相容的嗎？【不相容】試說(shuō)明。【RA C-Dans =2.2376e-001 4.7289e

12、-001-6.3633e-001 -3.0354e-001-1.7227e-001 -1.1938e-001-8.7955e-001 -6.5016e-001 C-Gans =2.2376e-001 4.7289e-001-6.3633e-001 -3.0354e-001-1.7227e-001 -1.1938e-001-8.7955e-001 -6.5016e-001 C-Hans =00000000000000001.3979e+000 1.3204e+0002.2485e-002 -1.5056e-0012.9484e-001 2.3624e-0018.0370e-002 -2.1506

13、e-0011.3979e+000 1.3204e+0002.2485e-002 -1.5056e-0012.9484e-001 2.3624e-0018.0370e-002 -2.1506e-001 D-G ans =0000000000000000 D-H ans =-2.2376e-001 -4.7289e-001 -1.3979e+000 -1.3204e+0006.3633e-001 3.0354e-001 -2.2485e-002 1.5056e-0011.7227e-001 1.1938e-001 -2.9484e-001 -2.3624e-0018.7955e-001 6.501

14、6e-001 -8.0370e-002 2.1506e-001 G-Hans =-2.2376e-001 -4.7289e-001 -1.3979e+000 -1.3204e+0006.3633e-001 3.0354e-001 -2.2485e-002 1.5056e-0018.7955e-001 6.5016e-001 -8.0370e-002 2.1506e-001(2) C=A*B; D=(A*B); G=B*A; H=(B*A); C-Dans =-2.2376e-001 6.3633e-001 1.7227e-001 8.7955e-001-4.7289e-001 3.0354e-

15、001 1.1938e-001 6.5016e-001-1.3979e+000 -2.2485e-002 -2.9484e-001 -8.0370e-002-1.3204e+000 1.5056e-001 -2.3624e-001 2.1506e-001 C-G ans =-2.2376e-001 6.3633e-001 1.7227e-001 8.7955e-001-4.7289e-001 3.0354e-001 1.1938e-001 6.5016e-001 C-H ans =0000000000000000 D-G ans =0000000000000000 D-H ans =1.397

16、9e+000 2.2485e-002 2.9484e-001 8.0370e-002 1.3204e+000 -1.5056e-001 2.3624e-001 -2.1506e-001 G-Hans =2.2376e-001 -6.3633e-001 -1.7227e-001 -8.7955e-0014.7289e-001 -3.0354e-001 -1.1938e-001 -6.5016e-0011.3979e+000 2.2485e-002 2.9484e-001 8.0370e-0021.3204e+000 -1.5056e-001 2.3624e-001 -2.1506e-001 C=

17、inv(A*B); D=inv(A)*inv(B); G=inv(B*A); H=inv(B)*inv(A); C-Dans =-3.9602e+001 -1.4016e+001 1.4537e+001 2.2261e+001 1.5266e+001 1.5778e+001 -1.9398e+001 -3.9304e+0011.3845e+001 -5.5182e-001 2.6289e+001 5.1120e+001 C-G ans =-3.9602e+001 -1.4016e+001 1.4537e+001 2.2261e+0011.5266e+001 1.5778e+001 -1.939

18、8e+001 -3.9304e+0011.0821e+001 1.4313e+000 -2.7296e+001 -4.8956e+0011.3845e+001 -5.5182e-0012.6289e+0015.1120e+0011.3845e+001 -5.5182e-0012.6289e+0015.1120e+001 C-Hans =-5.6843e-014 -1.2879e-0143.0198e-0147.1054e-014-6.5370e-013 -1.4744e-0133.3396e-0138.2423e-013-1.5774e-012 -3.5527e-0137.8870e-0131

19、.9895e-012-5.6843e-014 -1.2879e-0143.0198e-0147.1054e-014-6.5370e-013 -1.4744e-0133.3396e-0138.2423e-013-1.5774e-012 -3.5527e-0137.8870e-0131.9895e-0121.8758e-0124.2988e-013 -9.4502e-013 -2.4016e-0121.8758e-012 D-G ans =4.9738e-013 1.1013e-013 -8.3489e-014 -3.1264e-0131.7053e-013 3.7303e-014 -2.4869

20、e-014 -1.0747e-0135.8265e-013 1.3145e-013 -9.4147e-014 -3.8369e-013-1.0516e-012 -2.3448e-013 1.7053e-013 6.6791e-013 D-H ans =3.9602e+001 1.4016e+001 -1.4537e+001 -2.2261e+001-1.5266e+001 -1.5778e+001 1.9398e+001 3.9304e+001-1.0821e+001 -1.4313e+000 2.7296e+001 4.8956e+001-1.3845e+001 5.5182e-001 -2

21、.6289e+001 -5.1120e+001 G-H ans =3.9602e+001 1.4016e+001 -1.4537e+001 -2.2261e+001-1.5266e+001 -1.5778e+001 1.9398e+001 3.9304e+001-1.0821e+001 -1.4313e+000 2.7296e+001 4.8956e+001-1.3845e+001 5.5182e-001 -2.6289e+001 -5.1120e+001(4) c=inv(A*B); d=inv(A*B); g=inv(A)*inv(B); h=(inv(A)*inv(B); c-dans

22、=-3.9602e+001 1.5266e+001 1.0821e+001 1.3845e+001-1.4016e+001 1.5778e+001 1.4313e+000 -5.5182e-0011.4537e+001 -1.9398e+001 -2.7296e+001 2.6289e+0012.2261e+001 -3.9304e+001 -4.8956e+001 5.1120e+001 c-gans =-1.6875e-014 -5.4712e-013 -1.3216e-012 1.5774e-012-2.8866e-015 -1.3145e-013 -3.1264e-013 3.7659

23、e-0138.8818e-015 2.6290e-013 6.3949e-013 -7.6028e-0132.5757e-014 7.1765e-013 1.7195e-012 -2.0606e-012 c-hans =-3.9602e+001 1.5266e+001 1.0821e+001 1.3845e+001-1.4016e+001 1.5778e+001 1.4313e+000 -5.5182e-0011.4537e+001 -1.9398e+001 -2.7296e+001 2.6289e+0012.2261e+001 -3.9304e+001 -4.8956e+001 5.1120

24、e+001 d-g ans =3.9602e+001 -1.5266e+001 -1.0821e+001 -1.3845e+0011.4016e+001 -1.5778e+001 -1.4313e+000 5.5182e-001-1.4537e+001 1.9398e+001 2.7296e+001 -2.6289e+001-2.2261e+001 3.9304e+001 4.8956e+001 -5.1120e+001 d-h ans =-2.4158e-013 -1.1724e-013 -2.7711e-013 5.2580e-013-5.6843e-014 -1.8652e-014 -5

25、.3291e-014 1.0658e-0134.2633e-0141.5987e-0131.7764e-0146.7502e-0144.7962e-014 -8.8818e-0141.8474e-013 -3.3396e-013 g-hans =-3.9602e+0011.5266e+0011.0821e+0011.3845e+001-1.4016e+0011.5778e+0011.4313e+000 -5.5182e-0011.4537e+001-1.9398e+001 -2.7296e+0012.6289e+0012.2261e+001-3.9304e+001 -4.8956e+0015.

26、1120e+001第二題：(1) n=200; A=floor(10*rand(n); b=sum(A); z=ones(n,1); c=linsolve(A,b); d=c-z ;精度為 1e-141e-13;tic,x=Ab,toc=Elapsed time is 0.016000 seconds.tic,x=Ab,toctic,inv(A)*b,toc=Elapsed time is 0.031000 seconds.tic,inv(A)*b,toc(2)n=500; tic,x=Ab;tocElapsed time is 0.187000 seconds. 更快！ tic,y=inv(

27、A)*b;tocElapsed time is 0.343000 seconds. max(abs(x-z)=4.3987e-013更精確! max(abs(y-z) = 2.2524e-012 n=1000; tic,x=Ab;tocElapsed time is 0.920000 seconds. 更快！ tic,y=inv(A)*b;tocElapsed time is 1.404000 seconds. max(abs(x-z) =1.8221e-012更精確! max(abs(y-z) =2.0862e-011(3) A=floor(10*rand(6);B=A ; A(:,6)=-

28、sum(B(1:5,:)A =06770-2058470-2467433-2385833-2718394-2573288-28 x=ones(6,1); b=A*xb =000000 det(A)= 0 rref(A)ans =10000-101000-100100-100010-100001-10000003.2 A=floor(10*rand(6); B=A; A(:,6)=-sum(B(1:5,:); B=x*1:6B =123456123456123456123456123456123456 A*Bans =0000000000000000000000000000000000003.3

29、 A=floor(10*rand(6); B=A; A(:,6)=-sum(B(1:5,:); C=floor(10*rand(6); B=x*1:6; D=B+C; A*C-A*D ans =0000000000000000000000000000000000004題： B=eye(10)-triu(ones(10),1); C=inv(B); B(10,1)=-1/256; B=eye(10)-triu(ones(10),1); B(10,1)=-1/256;行最簡(jiǎn)形： rref(B)ans =100000000-256010000000-128001000000-64000100000-

30、32000010000-16000001000-8000000100-4000000010-2000000001-10000000000 d=B*C;C =1124816326412825601124816326412800112481632640001124816320000112481600000112480000001124000000011200000000110000000001行最簡(jiǎn)形： rref(d)ans =10000000000100000000001000000000010000000000100000000001000000000010000000000100000000

31、00100000000000 det(B) ans =0 rref(B c) ans =100000000-2560010000000-1280001000000-640000100000-320000010000-160000001000-80000000100-40000000010-20000000001-1000000000001第五題：(1) A=floor(20*rand(6); B=floor(20*rand(6,1)-10; x=ABx =0.32683-2.76131.67640.76772-0.33957-1.5678 C=A B; a=rref(C)1000000.326

32、83010000-2.76130010001.67630001000.76772000010-0.33957000001-1.5678 a(:,7)-xans =1.3288e-0079.5485e-006-4.8037e-006-2.7883e-0065.0983e-007-1.9237e-006(2) A=floor(20*rand(6); B=floor(20*rand(6,1)-10; A(:,3)=A(:,1:2)*4 3; rref(A B)ans =104000001300000001000000010000000100000001 y=floor(20*rand(6,1)-10

33、; A=floor(20*rand(6); A(:,3)=A(:,1:2)*4 3; c=A*y184139197147292 rref(A c) ans =104000101300060001008000010-100000160000000 A=floor(20*rand(6); A(:,3)=A(:,1:2)*4 3; y=floor(20*rand(6,1)-10; c=A*y;10400023013000170001009000010000000170000000 w=23,17,0,9,0,7; A*w-cans =000000(5) A=floor(20*rand(6); A(:

34、,3)=A(:,1:2)*4 3; U=rref(A)104000013000000100000010000001000000 Z=-4,-3,1,0,0,0; A*Zans =000000(6) A=floor(20*rand(6);A(:,3)=A(:,1:2)*4 3; y=floor(20*rand(6,1)-10; c=A*y;104000001300000001000000010000000100000000 U=rref(A c)U =1 U=rref(A c)U =104000-23013000-220001004000010900000160000000 w=-23,-22,

35、0,4,9,6; v=w+3*z; A*v-c ans =00000第六題:(1)A=0,1,0,1,0,0,0,1;1,0,1,0,0,0,1,0;0,1,0,1,0,0,0,0;1,0,1,0,1,0,0,0;0,0,0,1,0,1,0,0;0,0,0,0,1,0,1,0;0,1,0,0,0,1,0,1;1,0,0,0,0,0,1,0A =0101000110100010010100001010100000010100000010100100010110000010 AA2 ans =30201020030201022020101002030101101020100101020120101

36、03002010102 A1=AA2+AA1 =331211312212112131011010110011011211221121001101101129015101129015109113010018010718017171151071101201sum(sum(A1(1:8,1:8)ans =72 A2=AA4+AA2 = TOC o 1-5 h z 18113111811513110111511590710908151100113010 sum(sum(A2(1:8,1:8) ans =A3 =1191861640103111191103064186861631470730110319

37、31560736404713815600640561381471031730561931186073047163 sum(sum(A3(1:8,1:8) ans =2362 A4=AA8+AA4 =79915771436069711799169704361577577141813160501016971614138105014360316124313810043603811243131669715010381161411577050103161418 sum(sum(A4(1:8,1:8)ans =15800(4) A5=AA3+AA5 =080703068060307006060203706

38、0504003050402302040500704050660302060 sum(sum(A5(1:8,1:8) ans =A6 =047041024034470340240410034031017023410310240330024024016017240170160240041033024031340230170310 sum(sum(A6(1:8,1:8) ans =922 A7=AA7+AA7 =03090270016702233090223016702700022301970120015927001970150023200167015009501201670120095015000

39、2700232015001972230159012001970 sum(sum(A7(1:8,1:8) ans =6098 AA2 ans =3020102003020102202010100203010110102010010102012010103002010102 AA4 ans =1801309015001801509013130100701000150150801090707080090807071501008015001301007010 AA6 ans =119086064010300119010306408686063047073001030930560736404703805

40、600640560380471030730560930086073047063 AA8ans =79905770436069700799069704360577577041803160501006970614038105014360316024303810043603810243031669705010381061400577050103160418 AA3ans =0706030570503060050502036050404003040302302030400604040550302050 AA5 ans =04604002403346033024040003303001702340030

41、0230330024023015017240170150230040033023030330230170300 AA7030802690167022230802220167026900222019601200159269019601490232001670149094012016701200940149002690232014901962220159012001960 B=A; B(3,6)=1; B(6,3)=1; B(5,8)=1; B(8,5)=1; BA2 ans =tvS 0L0L090LL0L090Z00L0L0L09L0L0L090090L0L0L90L0L0L00L090L0L

42、L090L0Z0=SUBCvS BA5 ans =061061610610061061610610060061600610061060610600060061600610061060610600061061610610061061610610ans =18301820182018200183018201820182182018301820182001820183018201821820182018301820018201820183018218201820182018300182018201820183 BA7 ans =054705470546054754705470546054700547

43、05470547054654705470547054600546054705470547546054705470547005470546054705475470546054705470 BA8 ans =16410164001640016400016410164001640016401640016410164001640001640016410164001640164001640016410164000164001640016410164016400164001640016410164001641(6) B(6,8)=1; B(8,6)=1; C=B; CC =0101000110100010

44、010101001010100000010101001010110100010110001110 CA2 ans =302021200302020220302021020302022020312112021412202021310 CA3 ans =21212140717161870706272170718167070726216172828628284871716282882628784ans =222202211121822122042242220222221821112202214224222142142313221311228221331132621421422132313822112

45、213261331 CA5 ans =1263156321702175638638643265321563126321752170638638653264322164216530793079703275327960796821652164307930797532703279687960第七題： A=magic(8)A =64236160675795554121351501617474620214342244026273736303133323435292838392541232244451918484915145253111056858595462631 c=1:8; b=rref(A c)1

46、0011001001034-3-470001-3-445-70000000001000000000000000000000000000000000000(2) A=magic(8)A =64236160675795554121351501617474620214342244026273736303133323435292838392541232244451918484915145253111056858595462631 b=8 -8 -8 8 8 -8 -8 8; U=rref(A b)U =10011001001034-3-47-8001-3-445-7800000000000000000

47、0000000000000000000000000000 x2=floor(10*rand(5,1)x2 =67 c=U(1:3,9); V=U(1:3,4:8); x1=-1*V*x2+cx1 =-14-6-10 x=x1;x2; A*x-bans =000第八題：B=-1,-1;1,1; A=zeros(2),eye(2);eye(2),BA =0010000110-1-10111 c=B*B TOC o 1-5 h z 0000 AA2 ans =10-1-10111-1-1101101 AA4 ans =10 -2-20122-2-2 102201 AA6 ans =10-3-3013

48、3-3-3103301 AA8ans =10-4-40144-4-4104401 symsk C=eye(2),k*B;k*B,eye AA(2*K)的通項(xiàng)C =1,。, -k，-k0, 1, k, k-k, -k, 1, 0 k, k, 0, 1 C*(AA2) ans =1,0, -k-1, -k-10,1, k+1, k+1 -k-1, -k-1,1,0k+1, k+1,0,1(2) AA3ans = TOC o 1-5 h z -1-1 10110110 -2-20122 AA5ans =-2-210220110-3-30133 AA7 ans =-3-310330110-4-4014

49、4 AA9 ans =-4-410440110-5-50155D=(k-1)/2*B,eye(2);eye(2),(k+1)/2*BAA(2k-1)的通項(xiàng)-1/2*k+1/2,-1/2*k+1/2,1,01/2*k-1/2, 1/2*k-1/2,0,11,0, -1/2*k-1/2, -1/2*k-1/20,1, 1/2*k+1/2, 1/2*k+1/2 D*AA2 ans =-1/2*k-1/2, -1/2*k-1/2,1,0 1/2*k+1/2, 1/2*k+1/2,0,11,0,-3/2-1/2*k, -3/2-1/2*k0,1, 3/2+1/2*k, 3/2+1/2*k D=(k+1)

50、/2*B,eye(2);eye(2),(k+3)/2*BD =-1/2*k-1/2, -1/2*k-1/2,1,0 1/2*k+1/2, 1/2*k+1/2,0,11,0, -3/2-1/2*k, -3/2-1/2*k1,0,1, 3/2+1/2*k, 3/2+1/2*k第九題： A=floor(10*rand(6)863143531682281383656865873887045563 Aans =852680638574311635163885488686323573 B=A*AB =19316591156188126165199100164222134911008110911880156

51、16410919921813518822211821828015612613480135156105 B11=B(1:3,1:3); B12=B(1:3,4:6); B21=B(4:6,1:3); B22=B(4:6,4:6); C=inv(B11); C1=C0.018496-0.012892-0.0048636-0.0128920.022223-0.012953-0.0048636-0.0129530.033801 G=B21*CG =0.240980.221640.801320.0413510.981470.198640.21390.317320.35559 H=B22-B21*C*B21H =37.71428.93610.83128.93630.93.381210.8313.38127.08 L=eye(3),zeros(3);G,eye(3)1000000100000010000.240980.221640.801321000.0413510.981470.198640100.21390.317320.35559001 D= B11,zeros(3);zeros(3),H1931659100016519910000091100810000037.71428.93610.83100028.93630.93.381200010.8313.38127.0