矩阵与解线性方程组

20113No．32011JournalofChinesePeople'sPublicSecurityUniversityScienceandTechnology69Sum69100038。。。、。O241.61957—。1“”1。。1m×nA=a11a12…a1na21a22…a2n…………am1am2…amn1m×naijAi、jAA=aijA=aijm×nA=Am×n。m=nAn。2A、B1A、BA、BA=B。2A+B=aij+bijABA－B=aij－bijAB。3λA=λaij。3A=aijm×sB=bijs×ncij=∑sk=1aikbkji=12…mj=12…n2m×nC=cijm×nABC=AB。A、BABC。、。a11x1+a12x2+…+a1nxn=b1a21x1+a22x2+…+a2nxn=b2……………………………am1x1+am2x2+…+amnxn=bm．3A=a11a12…a1na21a22…a2n…………am1am2…amnX=x1x2xnb=b1b2bm．4A3X3b33AX=b．5·601·24AλAiA1Ai、jA2AiμjA3。A1、A2、A3A3。5A=aijm×nAkk、k×kkAk。ArArnkA=r。AnAn|A|aijn－1×n－1M－1i+jMaij。AA。6nE10E。nAnBAB=BA=EABA。AA－1。1nA|A|≠0A－1=1|A|A*A*=A11A21…An1A12A22…An2…………A1nA2n…Ann6AijAaij。。32AX=0A=Am×n。1rnkA=nAX=02rnkA=r＜nAX=0n－rn－r。r=mAmm0AXA=A1A2X=X1X()27A1m|A1|≠0A2m×n－mX1mX2n－m。X0=－A－11A2()E8n－mAX=0En－m。XAX=0AX=A1A2X1X()2=A1X1+A2X2=0X1=－A－11A2X2。X=X1X()2=－A－11A2X2X()2=－A－11A2()EX2=X0X2．X8X0X2n－m。Ar、rr0。AXA=A1A2A3A()4X=X1X()2A1r|A1|≠0A2r×n－rX1rX2n－r。AA*A=A1A2A3A()4→A*=A1A2()00．9AX=0A*X=0A*X=0A1A2X1X()2=0。3AX=brnkAb=rnkA。AbA。Ar、rr0A、X、bA=A1A2A3A()4X=X1X()2b=b1b()210A1r|A1|≠0A2r×n－rX1rX2n－rb1r·701·b2m－r。AbAb=A1A2b1A3A4b()2→A1A2b100()e．1111AX=bA1X1+A2X2=b1e=0rnkAb=rnkA。。4XbAX=bX0AX=0Xb+X0AX=bYb、XbAX=bX0=Yb－XbAX=0。AX=b。AXb+X0=AXb+AX0=b+0=bAYb－Xb=AYb－AXb=b－b=0．。x1－x2－2x3+2x4=－12x1+x2+x3－2x4=1x1+2x2+3x3－4x4=2{．12Ab1－1－22－1211－21123－42→1－1－22－1035－6300000rnkAb=rnkA=2124－2=2。12x1－x2－2x3+2x4=－13x2+5x3－6x4=3{．1313x3=0x4=0112Xb=0100．13x1－x2－2x3+2x4=03x2+5x3－6x4=0{．x3=1x4=0x3=0x4=12X1=1/3－5/310X2=0201．12。1312X=Xb+α1X1+α2X2α1、α2。、。。1．J．20101100．2．M．1978．3．M．1999．4．M．2005．·801·