00a0b000f00c0d0e??p1p2p3p4(?1)?(p1p2p3p4)ap11ap22ap33ap44观察行列式中元素0的位置及由4级排列中各数不能相等知,p1?3,p2?1,p3?4,p4?2,?(p1p2p3p4)??(3142)?3,0因此0a0b000f00c0d0e?(?1)a31a12a43a24??abcd3

[手段2]

习题3: 15913261014371115481216

解答: 159132610143711154111137111111?0

8c2?c1512c4?c391613115[技巧5,手段4] 习题4: 333x?3?3?3x?3?33x?333x?3?3?3?3

解答:

333x?31?x111?3?3x?3?3?3?3x?3?313x?3333x?333?3x0030x0x?3?3?3?3x?3?3?3?3r2?r1r3?r1xr4?r110004xc1??3?3x?3?3?30x03x003x?333x?3?x?x?xx?3?3?3?3?i?2cixxx

x?3?x?x?x?x4r2?r3?x000[技巧2,技巧3,技巧5,手段3]

习题5: a1b1a1b2a1b3a1b4a1b2a2b2a2b3a2b4a1b3a2b3a3b3a3b4a1b4a2b4a3b4a4b4

解答:

a1b1按第一列展开,a1b2a1b3a1b4a2b2?a1b1a2b3a2b4a2b3a3b3a3b4a1b2a2b2a2b3a2b4a1b3a2b3a3b3a3b4a1b2a2b4a1b4a2b4a3b4a4b4a1b3a3b3a3b4a1b4a4b4a2b4a4b4a1b2a2b4a1b3a2b3a3b4a1b4a4b4a1b2a2b3a1b3a2b3a3b3a1b4a2b4a3b4a3b4?a1b2a2b3a3b4?a1b3a2b2a2b4?a1b4a2b2a1b2a1b3a1b4a1b2a1b3a1b4由于行列式a2b2a2b3a2b4和a2b2a2b3a2b4有两行元素成比例,因此值为0,a2b4a3b4a4b4a2b3a3b3a3b4b2b3b4b2b3b4b2b3D?a1a2b1b4a2b3aa23b3a3b3a23b4?a1b2b4a2b3a3b4?(a1a2b1b4?1b2b4)a2b3a3b3a2a3a4a2a3a4a2a3?(a2221a2b1b4?a1b2b4)(a3a4b2b3?a2a3b3b4?a2a4b3?a3b2b4)?a1b4(a2b1?a1b2)[a4b3(a3b2?a2b3)?a3b4(a2b3?a3b2)]?a1b4(a2b1?a1b2)(a3b2?a2b3)(a4b3?a3b4)??a1b4(a1b2?a2b1)(a2b3?a3b2)(a3b4?a4b3)3??a1b4?(aibi?1?ai?1bi)i?1[技巧7,手段1,手段6] 习题6: a4(a?1)4(a?2)4(a?3)4(a?4)4a3(a?1)3(a?2)3(a?3)3(a?4)3a2(a?1)2(a?2)2(a?3)2(a?4)2

aa?1a?2a?3a?411111解答:

b4a3b4a4将行列式上下翻转后再左右翻转,不难得1aD5?(?1)4?3?2?12341a?1(a?1)(a?1)(a?1)2341a?2(a?2)(a?2)(a?2)1a?32342342341a?3(a?3)(a?3)(a?3)1a?2(a?2)(a?2)(a?2)2342341a?4(a?4)(a?4)(a?4)1a?1(a?1)(a?1)(a?1)234234aaa 1aaaa2341a?4?(?1)4?3?2?14?3?2?1(?1)(a?4)(a?4)(a?4)(a?3)(a?3)(a?3)?4!3!2!1!?288[技巧2,手段7]

习题7:

x00?0an?1x0?0an?10?1x?0an?2?????000?xa2000??1x?a1

解答:

按第一列展开,得Dn的递推公式?1xDn?xDn?1?an(?1)n?10?000?000??1?1?x?0??0?0x?Dn?xDn?1?anDn?1?xDn?2?an?1Dn?2?xDn?3?an?2?D2?xD1?a2D1?x?a1将上述各式的两边分别乘以1,x,x,?,xDn?x?a1xnn?12n?1

后全部相加,并化简,得:?a2xn?2???an?1x?an[技巧7,手段3] 习题8:

a?ac?cbd?d?(其余未写出元素均为零): b解答: 将D2n中的第2n行依此与第2n?1行,2n?3行,?,第2行对调,再将第2n列依此与第2n?1列,2n?3列,?,第2列对调,得ac02(2n?2)bd000a???00b?D?(?1)2na??c?00c2(n?1)b

d?d?D2D2(n?1)?(ad?bc)D???(ad?bc)n?1D2?(ad?bc)n[技巧2,技巧6]