(2) -5b 2.5 b 命题1 设A为m阶可逆矩阵\ue003B为m×n矩阵\ue003试证R(AB)=R(B)AR( A)siAAB即PP\ue005P ,2是 经过 次初等行变换得出的,P(i, ,,s)\ue005\ue005121 2其中皆为初等矩阵,sP PABR AB)P B,BsR(B).\ue0051则(证因A为m阶可逆矩阵\ue003,B,R(B),AB,R( AB).1 111 1 10\ue0051 1 11 1 1201021\ue00e\ue010\ue015\ue017\ue00e\ue010\ue015\ue017\ue00e\ue010\ue015\ue017\ue005\ue005\ue005\ue005\ue00f\ue016\ue00f\ue016\ue00f\ue016\ue005例 有关矩阵秩的若干常用结论 命题2 设A为n阶可逆矩阵\ue003B为m×n矩阵\ue003试证R(BA)=R(B)11 11111因A为n阶可逆矩阵\ue003siABA即PPBP PBAR BA)P ,P(i, ,,s)\ue005\ue005121 2其中皆为初等矩阵,sP ,BsR(B).\ue005\ue00512是 经过 次初等列变换得出的,则(B,A,BAR(B ),R( A),R(BA).0001 11 11 1122\ue00e\ue00f\ue010\ue015\ue016\ue017\ue00e\ue00f\ue010\ue015\ue016\ue017\ue00e\ue010\ue015\ue017\ue00f\ue00f\ue016\ue016\ue00f\ue00f\ue016\ue016\ue005\ue005\ue005\ue00f\ue016\ue005\ue005\ue005例 证 m pEp nR( AB),B,R(B)\ue00e\ue00f\ue00f\ue010\ue005p3 AMm tAB0\ue005\ue015\ue017\ue005Bt n0,R( M )1 11p\ue00b\ue00b\ue00b\ue00b\ue00e\ue00f\ue015\ue016\ue005 \ue00f\ue016\ue017\ue010\ue002命题设 为矩阵, 为矩阵以及则A,AB,ER( A),R( AB)\ue015\ue016\ue016\ue0172.21 1 10\ue0051 1 11 1 11 00 1201021\ue00e\ue010\ue00e\ue010\ue015\ue017\ue00e\ue010\ue015\ue017\ue00e\ue010\ue015\ue017\ue005\ue005\ue005\ue0051 1 10\ue005\ue00f\ue016\ue00f\ue016\ue00f\ue016\ue00f\ue016例 AB0MER( M )R( AB)\ue00e\ue00f\ue00f\ue010\ue015\ue016\ue016\ue017\ue00e\ue00f\ue00f\ue010\ue015\ue016\ue016\ue017\ue00f\ue00f\ue00f\ue016\ue016\ue016\ue00f\ue00f\ue00f\ue016\ue016\ue016\ue00f\ue00f\ue00f\ue016\ue016\ue016\ue00e\ue010\ue015\ue017\ue005\ue005\ue005\ue009\ue009\ue00f\ue016\ue00221 1 11 1 11 1 100010 1010 1010 100003 ss ( n s )( m s ) s( m s ) ( n s )sps p( m s ) ( n s )sR( AB)E0s,00M\ue003\ue00bMABEEE,0R( M )pR( AB)p\ue00b\ue003\ue00b\ue003\ue003\ue002\ue003\ue005 \ue002\ue00b\ue003\ue005\ue00e\ue00f\ue010\ue015\ue016\ue016\ue00f\ue00f\ue016\ue016\ue017\ue009 \ue00f\ue00e\ue00f\ue015\ue016\ue009 \ue00f\ue016\ue017\ue010\ue005\ue002证 设对作初等变换,,其中将化为标准形,则 m nBtA\ue005\ue00fm tAER( A)Bt n0C,R(C )R(B)\ue00b\ue00b\ue00b\ue00e\ue010\ue00a1 1 1011 1110 13\ue015\ue017\ue016\ue002\ue015\ue017命题4设 为矩阵, 为矩阵以及则AR( A),B,E,R(B )\ue00e\ue015\ue00f\ue00f\ue005.21 11\ue00510012200\ue00e\ue010\ue015\ue017\ue00e\ue010\ue005\ue00e\ue010\ue015\ue017\ue005\ue005\ue005\ue00f\ue016\ue00f\ue016\ue00f\ue016例A