《结构工程中的数学方法》课程教学课件（讲稿）matrix_inversion

2AlgebraicproblemsinmatrixformSystemsofLinearEquationsMatrix inversion·Aand B are nxn matrices andB so that A·B=B.A=InwithI, =identitymatrix,then B =A-1 (inverse of A)[AailA12Ana12ainarA...A1A2...A..Aza21a22a,ia2n目1：：：AB=A-1det(A)Ai1..A..A..anAza2dia.目1：::.:A.aniA.A.....A..an2aannAi:elements of the adjugate ofmatrixA (cofactors)A, = CjC, = (-1)*. M,Mi:(i,j)-minor ofA,Mj=det(A_i-j)with Ai-iobtained from matrixAbydeletingrowiand columnj(A_i-j is an (n-1)×(n-1) matrix)52aMichael Beer,Engineering Mathematics

Michael Beer, Engineering Mathematics A and B are nxn matrices and $ B so that A·B = B·A = In, with In = identity matrix, then B = A−1 (inverse of A) Systems of Linear Equations Matrix inversion ● 11 12 1j 1n 21 22 2 j 2n i1 i2 ij in n1 n2 nj nn a a . a . a a a . a . a a a . a . a a a . a . a     =   A       2 Algebraic problems in matrix form 52a ( ) 11 12 1j 1n 21 22 2 j 2n 1 i1 i2 ij in n1 n2 nj nn A A . A . A A A . A . A 1 det A A . A . A A A . A . A −     = = ⋅   B A A       A C ij ji = Aij: elements of the adjugate of matrix A (cofactors) ( ) i j C 1M ij ij + =− ⋅ Mij: (i,j)-minor of A, Mij= det(A−i−j ) with A−i−j obtained from matrix A by deleting row i and column j (A−i−j is an (n−1)×(n−1) matrix)