《線性代數(shù)》行列式討論題解答

1、《線性代數(shù)》第一章《線性代數(shù)》第一章行列式行列式討論題討論題1.計算行列式12125431432321??nnnD?????????[解法一]：后行減前行；每列減第一列；第一列乘以n各列加到第一列；按第一列展開1111111111111111321?????????nnnnDn????00010001000112211?????????nnnnn?????00000000012211?????????nnnnnnnnnn?????000

2、000000000122111?????????nnnnninni???????2)1(1))112)1)(2(??????nnnn((nnn12)1(2)1()1nnnnn(????[解法二]：各列加到第一列；后行減前行；每列減第一列；第一列乘以n各列加到第一列；按第一列展開1211254114313212)1(???nnnnD?????????11110111101111013212)1(?????????nnnnnnn?????1

3、0000000111112)1(???????nnnnnnnnnn????????0000000000123212)1(????????nnnnnn???????2)1()1()1(212)1)(2(????????nnnnnnn12)1(2)1()1nnnnn(????2.用遞推法計算（提示：表示元集的組合數(shù)，利用nnnnnnnxCCCCxCCxCD13212221211110010001100001??????????????knC

4、nk）。111?????knknknCCC解：后行減前行，得遞推公式（）111kkknnnCCC?????=11112212121122331111111100001000010000nnnnnnnnnnnnnCxDCCCxxCCCCCCCCxx????????????????????????????1112211111000010100010100(1)01(1)nnnnnxCxxCCCxx????????????????????(1

5、)nxD?[解法一]拆列，共有個行列式，n2時，；.2n0nD?1112122112()()nDabnDaabb???????[解法二]利用矩陣乘積1112111232122221210010011111000000nnnnnnnnnababababbbbabababaDabababa????????????????????????????????????????????????????????112321001001111010000

6、00nnabbbbaa?????????????????6計算行列式按第一行展開44332211500000000000011111xaxaxaxaD?????1122334411111000000000000axaxaxax?????4433221100000000000011111xaxaxaxa?????4433221100000000000011111xaxaxaxa?????4433221100000000000011111x

7、axaxaxa?????4433221100000000000011111xaxaxaxa?????=43214321432143214321aaaaxaaaxxaaxxxaxxxx????本題推廣到一般情形:nnnnnnxaaaxxaaxxaxxxD1213212121???????????7.證明=xaxaxaxaxaxaxaxaxaxaxaxaxaxaxaxa????????????????44434241343332312423

8、222114131211???4144434241343332312423222114131211jiijAxaaaaaaaaaaaaaaaa其中為等式右邊第一個行列式中元素的代數(shù)余子式。本題推廣到一般情形，應(yīng)該是什么結(jié)果？ijAija證法一：加邊:444342413433323124232221141312114443424134333231242322211413121141111100001aaaaaaaaaaaaaxaxaxax