解 系数行列式为
?11 D?1?1??????
12?1 令D?0? 得 ??0或??1?
于是? 当??0或??1时该齐次线性方程组有非零解?
??(1??)x1?2x2?4x3?0 10? 问?取何值时? 齐次线性方程组?2x1?(3??)x2?x3?0??x1?x2?(1??)x3?0有非零解？
解 系数行列式为
1???241???3??4 D?23??1?21??1
111??101?? ?(1??)3?(??3)?4(1??)?2(1??)(?3??) ?(1??)3?2(1??)2???3? 令D?0? 得
??0? ??2或??3?
于是? 当??0? ??2或??3时? 该齐次线性方程组有非零解?
第二章 矩阵及其运算
1? 已知线性变换?
??x1?2y1?2y2?y3?x2?3y1?y2?5y3? ??x3?3y1?2y2?3y3求从变量x1? x2? x3到变量y1? y2? y3的线性变换? 解 由已知?
?x1??221??y1? ?x2???315??y2??
?x??323??y???2??3???y1??221??x1???7?49??y1?故 ?y2???315??x2???63?7??y2??
?y??323??x??32?4?????3????y3??2???1??y1??7x1?4x2?9x3 ?y2?6x1?3x2?7x3?
??y3?3x1?2x2?4x3 2? 已知两个线性变换
??x1?2y1?y3 ?x2??2y1?3y2?2y3?
??x3?4y1?y2?5y3 解 由已知
??y1??3z1?z2?y2?2z1?z3? ??y3??z2?3z3求从z1? z2? z3到x1? x2? x3的线性变换?
?x1??201??y1??201???31 ?x2????232??y2????232??20?x??415??y??415??0?1??2?????3????613??z1? ??12?49??z2??
??10?116??z????3?0??z1?1??z2? ??3???z3???x1??6z1?z2?3z3所以有?x2?12z1?4z2?9z3?
??x3??10z1?z2?16z3?111??123? 3? 设A??11?1?? B???1?24?? 求3AB?2A及ATB?
?1?11??051??????111??123??111? 解 3AB?2A?3?11?1???1?24??2?11?1?
?1?11??051??1?11????????058??111???21322? ?3?0?56??2?11?1????2?1720??
?290??1?11??429?2????????111??123??058? ATB??11?1???1?24???0?56??
?1?11??051??290??????? 4? 计算下列乘积?
?431??7? (1)?1?23??2??
?570??1??????431??7??4?7?3?2?1?1??35? 解 ?1?23??2???1?7?(?2)?2?3?1???6??
?570??1??5?7?7?2?0?1??49??????????3? (2)(123)?2??
?1????3? 解 (123)?2??(1?3?2?2?3?1)?(10)?
?1????2? (3)?1?(?12)?
?3????2?(?1)2?2???2?2? 解 ?1?(?12)??1?(?1)1?2????1?3??3?(?1)3?2???3??????1?02140?? (4)???11?134????43?1?301?2? ? 1??2??4?2?? 6???1?02140?? 解 ???11?134????43?1?301?2??6?78??
?1???20?5?6????2??a11a12a13??x1? (5)(x1x2x3)?a12a22a23??x2??
????aaa?132333??x3? 解
?a11a12a13??x1? (x1x2x3)?a12a22a23??x2?
????aaa?132333??x3??x1? ?(a11x1?a12x2?a13x3 a12x1?a22x2?a23x3 a13x1?a23x2?a33x3)?x2?
?x??3?222 ?a11x1?a22x2?a33x3?2a12x1x2?2a13x1x3?2a23x2x3?
1 5? 设A???1?2?? B??10?? 问?
?12?3???? (1)AB?BA吗?
解 AB?BA?
3 因为AB???4?4?? BA??12?? 所以AB?BA?
?38?6???? (2)(A?B)2?A2?2AB?B2吗? 解 (A?B)2?A2?2AB?B2?
2 因为A?B???2?2?? 5??2 (A?B)2???2?2??2?25???2???814??
?1429?5????38??68???10???1016?? 但 A2?2AB?B2???411???812??34??1527?????????所以(A?B)2?A2?2AB?B2? (3)(A?B)(A?B)?A2?B2吗? 解 (A?B)(A?B)?A2?B2?
2 因为A?B???2?2?? A?B??0?05???2??0?05???2??
1??6??
9??2 (A?B)(A?B)???2?2???0?01???38???10???2而 A2?B2???411??34??1?????故(A?B)(A?B)?A2?B2?
8??
7?? 6? 举反列说明下列命题是错误的? (1)若A2?0? 则A?0?
0 解 取A???0?1?? 则A2?0? 但A?0? 0??