行列式化简计算技巧实题

行列式化简计算技巧和实题操练——Zachary一.技巧:技巧1:行列式与它的转置行列式的值相等,即D=DT111211121121222122221212nnnnnnnnnnnnaaaaaaaaaaaaaaaaaa技巧2:互换行列式的任意两行(列),行列式的值将改变正负号111212122221222111211212nnnnnnnnnnnnaaaaaaaaaaaaaaaaaa技巧3:行列式中某一行(列)的所有元素的公因子可以提到行列式记号的外面1111121111121221222222122211212nnnnnninnnnnnnnnnnbababaaaabababaaaabbababaaaa技巧4:行列式具有分行(列)相加性11121111211112111221212121212nnntttttntntttntttnnnnnnnnnnnnnaaaaaaaaabcbcbcbbbcccaaaaaaaaa技巧5:将行列式的某一行(列)的各元素乘以同一数k后加到另一行(列)对应的元素上,行列式的值不变111211112112112212121212nnsssnststsntntttntttnnnnnnnnnaaaaaaaaaakaakaakaaaaaaaaaaaaa技巧6:分块行列式的值等于其主对角线上两个子块行列式的值的乘积111111111111111111110000mmnmmmmnmmmnnnnnmnnnaaaabbaaccbbaabbccbb技巧7:[拉普拉斯按一行(列)展开定理]行列式等于它的任一行(列)的各元素与其对应的代数余子式乘积之和11(1,2,,)(1,2,,)nnikikkjkjkkDaAinaAjn二.解题方法:方法1:对于2阶行列式和3阶行列式,可以直接使用对角线法则进行计算1112112212212122aaaaaaaa,111213212223112233122331132132112332122133132231313233aaaaaaaaaaaaaaaaaaaaaaaaaaa方法2:上三角行列式,下三角行列式,主对角线行列式,副对角线行列式111212221000nnniiinnaaaaaaa,112122112000niiinnnnaaaaaaa,1212()nn其余未写出元素均为零,1(1)2212(1)()nnnn其余未写出元素均为零方法3:若行列式中有两行(列)对应元素相等,则此行列式的值等于零0aaeibbfjccgkddhl方法4:若行列式中有一行(列)的元素全为零,则此行列式的值为零00000aeibfjcgkdhl方法5:若行列式中有两行(列)元素成比例,则此行列式的值等于零0akaeibkbfjckcgkdkdhl实题操练：计算下列行列式的值:习题1:120114318解答:1201141182(4)30(1)(1)0132(1)81(4)(1)4318习题3:12345678910111213141516解答:21431234113156785171091011129111113141516131151cccc习题4:3333333333333333xxxx解答:412213141423333333333333333333333333333313331333133300133300133300013330000000iixxxxxxccxxxxxxxrrxxxxrrxxxxrrxxxxrrxxxxx习题5:11121314122223241323333414243444abababababababababababababababab解答:1112131412222324132333341424344422232412131412131411233334122333341322232414243444243444243444,abababababababababababababababababababababababababaabababababababababababababababababababababab按第一列展开1213142223242333341213141213142223242223242434442333342342342121423333412423333412234234,0,(bababababababababababababababababababababababababababbbbbbbDaabbababababbabababaaaaaaaa由于行列式和有两行元素成比例因此值为3234214124233334234222121412434232334243241421124332233423321421123223433414122123)()()()[()()]()()()()(bbbbbabbabababaaaaabbabbaabbaabbaababbabababababababababababababababababababab323443314111)()()iiiiiabababababab习题6:444443333322222(1)(2)(3)(4)(1)(2)(3)(4)(1)(2)(3)(4)123411111aaaaaaaaaaaaaaaaaaaa解答:432122222533333444444321432122222,111111234(1)(1)(2)(3)(4)(1)(2)(3)(4)(1)(2)(3)(4)111114321(1)(1)(4)(3)(2)(1)(4)aaaaaDaaaaaaaaaaaaaaaaaaaaaaaaaa将行列式上下翻转后再左右翻转不难得3333344444(3)(2)(1)(4)(3)(2)(1)4!3!2!1!288aaaaaaaaa习题7:12211000010000000001nnnxxxxaaaaxa解答:111121232212112112121,1000100(1)0000011,,,,,,nnnnnnnnnnnnnnnnnnnnnDxDxDaxxDxDaDxDaDxDaDxDaDxaxxxDxaxaxaxa按第一列展开得的递推公式将上述各式的两边分别乘以后全部相加并化简得:习题8:()ababcdcd其余未写出元素均为零:解答:22(22)2122(1)2(1)2221,23,,2,221,23,,2,000000(1)00()()()nnnnnnnDnnnnnnabcdabDabcdcdDDadbcDadbcDadbc将中的第行依此与第行行第行对调再将第列依此与第列列第列对调得