《线性代数》复习题
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a?1001b?1001c?100? 1d
01c?10r?ar01?ab120??1b????10?1d00a1c?101?aba00 ?(?1)(?1)2?1?1c1 10?1dda 解 ?1001b?10
c3?dc21?abaad??????1c1?cd0?10abad ?(?1)(?1)3?21??11?cd?abcd?ab?cd?ad?1?
＃. 用Gauss消元法解下列线性方程组.
?x1?2x2?7x3??4??2x1?x2?x3?13?3x?9x?36x??33231)?1
解: 1) 对增广矩阵进行变换:
?3x4?x1?2x2?x3?x4??3x2?2x4??2x?x2?4x32)?1?4?0?1?5
?12?7?4?r1?(?2)?r2?12?7?21??r?0?3151?(?3)?r3113???????????39?36?33???03?15r2?1?r3?12?7?4??1r3?(?1/3)?r2?(?2)?r1????????01?5?7?????????0??0??000??0则x3为自由变量, 令x3=t为任意实数, 则x1=10-3t, x2=5t-7, 方程有无穷多解, 解集为
(10-3t, 5t-7, t).
2) 对增广矩阵进行变换:
034?034? ?10?10??02? ?02?1?10?1?10r?(?2)?r14????????? ??03??0?21030?21? ????05?0?14?6?3 ?2?14?? r2?r434??100034??10 r?(?1)r2?3?r3??01?46?2?30?14?6?3r?(1/12)r?2?r 324??????????????52??001 ????0012?20?8?33????
007?13?6007?13?6??????
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r4?(?) 45 r4?()?r3 34??1003 21?r?(2)?r??10001?42r3?4?r2?010?33?3?01001?r?(?3)?r1 r3?(?7)?r4??41?52????????????? ?001????00101?33?????000?4?4??00011??33????4?21???21??0310?1?5?7??000??
方程有唯一解x1=x2=x3=x4=1.
＃. 确定下列线性方程组中k的值满足所要求的解的个数. 1) 无解: 2) 有唯一解:
3) 有无穷多解:
?x?2y?kz?6??3x?6y?8z?4;
?x?y?kz?4??x?2y?z?5?x?2y?z?1?
?kx?y?14??2x?3y??12
解:
1) 对增广矩阵作变换:
因此, 要使方程组无解, 须使8-3k=0, 解得k=8/3, 即当k取值为8/3时, 方程无解. 2) 对增广矩阵作变换:
k6??12k6?r1?(?3)?r2?12???????3684??008?3k?14?????
kr?(??3?12?14?r1?r2?2?3?12?12)?r2?2?k13k?????????????2?3?12?????0?16k?14k114??????2??
23kk???1?03. 因此, 如要方程组有唯一解, 必须有2, 即
3) 对增广矩阵作变换
k?11k4?r1?(?1)?r2?11?1215??r?011?k1?(?1)?r3???????????1?211???0?31?k4?k?11r2?3?r3?1????????011?k??3???004?4k因此, 如要方程组有无穷多解, 必须4-4k=0, 即当k=1时, 方程组才有无穷多解.
＃. 讨论以下述阶梯矩阵为增广矩阵的线性方程组是否有解; 如有解区分是唯一解还是无穷多解.
4?1??0??
解: 1) 方程组有一个自由变元x2, 因此方程组有无穷多解. 2) 方程组的三个变元均为首项变元, 因此方程组有唯一解. 3) 第三个方程0=4说明此方程无解.
4) 方程组的三个变元均为首项变元, 因此方程组有唯一解.
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??12?30??002?3????0000?? 1)??1?204??002?3????0004???0000? 3)?
?1?32?1??0203????0014?? 2)??1?20?1??0231????0010???0000? 4)?
＃. 对给定方程组的增广矩阵施行行初等变换求解线性方程组..
??3x?5y??22??3x?4y?4?4x?12y?7z?20w?22??x?8y?321)? 2)?3x?9y?5z?28w?30 ?2x?y?z?w?1?1111x?y?z?w??2222?4x?2y?2z?2w?2 3)?解: 1) 对增广矩阵进行变换:
方程组无解.
2) 对增广矩阵进行变换
??35?22??1?832?r1?(?3)?r2?1?832??3??r?3??r?028?92?3?r11?(3)?r344????44???????????????1?832????35?22???0?1974?????1?832??1?832?23?r2?19?r3?23?r2?(1/28)????????01?????????01??7?7???0?197481????00?7???
?412?7?2022?r1?(1/4)?13?74?39?5?2830????????39?5????1r1?(?3)?r2?????????0?r2?4/7?r1?1????????03?11?2??2830???511?2??5254???5
可以看出y和w为自由变元, 则令y=s, w=t, s与t为任意常数, 则x=100-3s+96t, z=54+52t. 方程的解集表示为(100-3s+96t, s, 54+52t, t). 3) 对增广矩阵进行变换
711?7?5??r?413?42??2????4127??0?13?00142?30?96100?01?5254???2??1?4?1122?11?2?211?22?1??11?r1?r2??????2?2???42???12001?2001?2241212?12?1?2?12121?2?1??2???1200?10201001?2?0??0???
?r1?(?2)?r2?1r1?(?4)?r3???????0??0??1?r2?(?2)?r3?1r2??1/2???2r2?(1/2)?r1?0????????0??0??0????可知y与z为自由变元, 令y=s, z=t, s与t均为任意实数, 则
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x?1?11?111??s?t,s,t,0??s?t,w?02? 222, 方程组的解集为?22
＃. 对给定齐次线性方程组的系数矩阵施行行初等变换求解下列方程组.
解:对系数矩阵作初等变换
?x?2y?z?w?0??x?2y?2w?0?2y?z?w?0?
因此, w为自由变元, 令w=t为任意实数, 则x=-2t, y=0, z=t, 方程组的解集为 (2t, 0, t, t).
＃. 设一线性方程组的增广矩阵为
11?11?11??12?12?12?1?202??r?0?4?11??r?02?11?2?r31?(?1)?r2???????????????????02?11???02?11???0?4?11??r2?(?1)?r1?1020?r2?(1/2)20??10r2?2?r3??r?01?1/21/2?3?(?1/3)???????02?11???????????1?1??00?33???00?r3?(1/2)?r2?1002?r3?(?2)?r1?????????0100????001?1??
求α的值使得此方程组有唯一解.
解: 对增方矩阵求初等变换
211??1??1432?????2?2?3??
因此, 此方程组要有唯一解, 就必须满足α+2≠0, 即α≠-2.
＃. 设一线性方程组的增广矩阵为
211?r1?r211?11??1?12?12??1432??r?06??r?06?1?(?2)?r32?r3?????43???43??????????2?2?3???0?6??21???00??24??
1) 此方程有可能无解吗? 说明你的理由. 2) β取何值时方程组有无穷多解?
解: 1) 此方程一定有解, 因为此方程是齐次方程, 至少有零解. 3) 对此增广矩阵做初等变换
2?10??1??2?530?????0???14?
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2?10?r1?2?r2?12?10??10??1?12??2?530??r?0?1??r?0?1?1?r32?6?r3????10????10??????????0???14??06??10???00??50??
因此, 只有当β+5=0, 即β=-5时,方程才有无穷多解.
#. 求λ的值使得下述方程组有非零解.
解: 对系数矩阵作初等行变换:
y?0?(??2)x???x?(??2)y?0 ?因此, 要使方程有非零解, 必须有(λ-2)2+1=0, 但(λ-2)2+1≥0对λ取任何实数值总是成立, 因此必有(λ-2)2+1≠0, 因此, 无论λ取什么值此方程组都不会有非零解. ＃. 设
1?r1?r2??1??2?r1?(??2)?r2??1??2????2??????????????1??2????2?0(??2)2?1?1???????
?121?A????212?, ?432?B?????21?2?
求: 1) 3A-2B;
2) 若X满足AT+XT=BT, 求X.. 解: 1)
2)
因X满足AT+XT=BT, 等号两边同时转置, 有 A+X=B,
等号两边同时减去A, 得 X=B-A, 因此有
?123A?2B?3??21?3?8???6?(?4)1??432??3?2??21?2???62?????6?63?4???5???3?26?(?4)??1063??864????36??42?4???0?1?110??
＃. 计算下列矩阵的乘积:
?432??121??4?13?22?1?X?B?A?????212????2?21?1?2?2??21?2???????311??????40?4?
?3????121??1????2??; 1)
?12?123????212??01???30?3)
解:
1)
?1??2?????12??3???2) ?4?; 0??101??1??312?????1?1?02?1??????031?????1??; ??110????0?? 4)
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