第四章 线性方程组
§4-1 克拉默法则
一、选择题
1.下列说法正确的是( C )
A.n元齐次线性方程组必有n组解; B.n元齐次线性方程组必有n?1组解;
C.n元齐次线性方程组至少有一组解,即零解;
D.n元齐次线性方程组除了零解外,再也没有其他解. 2.下列说法错误的是( B )
A.当D?0时,非齐次线性方程组只有唯一解; B.当D?0时,非齐次线性方程组有无穷多解; C.若非齐次线性方程组至少有两个不同的解,则D?0; D.若非齐次线性方程组有无解,则D?0. 二、填空题
??x1?x2?x3?0?1.已知齐次线性方程组?x1??x2?x3?0有非零解,
?x?2?x?x?023?1则?? 1 ,?? 0 .
2.由克拉默法则可知,如果非齐次线性方程组的系数行列式D?0, 则方程组有唯一解xi? Di . D三、用克拉默法则求解下列方程组
?8x?3y?21.?
6x?2y?3?解:
D?8362??2?0
D1?2332??5,
D2?8263??12
所以,x?D15D?,y?2??6 D2D?x1?2x2?x3??2?2.?2x1?x2?3x3?1??x?x?x?0?123
1D?2解:
?2111?11?11?3?1r2?2r1r3?r11?2030?11?5??5?00
?1?2?200?501?150?51?3??10, ,
D1?101D2?2?11D3?2?111?210?3r1?2r211?3??5?3r1?2r225?10?10011??5
?2?21101r1?2r22?110所以, x1?DD1D?1,x2?2?2,x3?3?1DDD
?2x?z?1? 3.?2x?4y?z?1
??x?8y?3z?2?2D?2解:
0?1300?15810028521010?0, ,
4?1c1?2c304?1?20?03
?1810?128231?13D1?14?1c3?c1140?20D2?22D3?21?1c3?c22010?12?1250141c1?2c3041?20
?182所以, x??582DD1D?1,y?2?0,z?3?1 DDD?x1?x2?x3?x4?5?x?2x?x?4x??2?12344.?
?2x1?3x2?x3?5x4??2??3x1?x2?2x3?11x4?0解:
1111r2?r112?14D?r3?2r12?3?1?5r4?3r131211111101?230?5?3?70?2?18
1?231?23r2?5r1??5?3?70?138??142r?2r1?2?1830?5145111?22?14c3?2c2D1??2?3?1?5c4?11c20121151?1?10?22?5?18?2?35280100
5?1?100?10c1?5c2??2?5?18?27?332??142c3?10c2?2528235?221511r2?r11?2?14D2?r3?2r12?2?1?5r?3r130211415110?7?230?12?3?70?15?18?7?23230?13r1?2r3??12?3?7330?31??284r?3r3?15?182?15?18115112?24c1?3c2D3?2?3?2?5c4?11c231011?215?10?52?2?1811?3?2280100?25?10?2002c2?5c11??5?2?18?5?297??426c?5c1211?22831151?27
线性代数练习册习题及答案本
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