2022考研数学满分过关1501第一章行列式重点题型行列式的计算【例1】设12312,,,,均为4维列向量,1231(,,,)A=,3122(,,,)B=.若1A=,2B=,则2AB−=.【详解一】因为132132122(2,2,2,2)AB−=−−−−,则有1321321132132222,2,2,2,2,2,2AB−=−−−−−−−,又1321321123112000120(2,2,2,)(,,,)20100001−−−−−=−,于是有1321321120001202,2,2,20100001A−−−−−=−7=−.1321322312220101200(2,2,2,2)(,,,)01200002−−−−−=−,有1321322201012002,2,2,22801200002B−−−−−==−−,所以27(28)21AB−=−−−=.【详解二】(特值法)令1000010000100001A=,0100001010000002B=则V研客2022考研数学满分过关15021200012022120100003AB−−−==−−第二章矩阵重点题型一求高次幂【例2】设A为3阶矩阵,(9,18,18)Tb=−,非齐次线性方程组Axb=的通解为12(2,1,0)(2,0,1)(1,2,2)TTTkk−++−,其中12,kk为任意常数,求A与100A.【详解】由2100A−=,2001A=知12(2,1,0),(2,0,1)TT=−=为矩阵A属于特征值120==的两个线性无关的特征向量.由191218922182A==−−−知3(1,2,2)T=−为矩阵A属于特征值39=的特征向量.令123(,,)P=,则1009PAP−==得122102541221102024524490129122244APP−−−−===−−−−−故V研客2022考研数学满分过关1503100100199100221025412211020245924490129122244APP−−−−===−−−−−或9910099122()9244244AtrAA−==−−−重点题型二逆的判定与计算【例3】设,AB为n阶矩阵.(I)若A与ABE−可逆,证明BAE−可逆;(II)若ABE−可逆,证明BAE−可逆;【证明】(I)方法一:11111()||||()0EBAAABAABAABAAABAABEAB−−−−−−=−=−=−=−=−=−故BAE−可逆.方法二:因为A可逆,1()AABABA−=,ABBA,~EABEBA−−,ABE...