let n be Nat; :: thesis: for K being Field
for A being Matrix of n,K st Det A <> 0. K holds
for x, b being Matrix of K st len x = n & A * x = b holds
( x = (A ~ ) * b & ( for i, j being Nat st [i,j] in Indices x holds
x * i,j = ((Det A) " ) * (Det (ReplaceCol A,i,(Col b,j))) ) )
let K be Field; :: thesis: for A being Matrix of n,K st Det A <> 0. K holds
for x, b being Matrix of K st len x = n & A * x = b holds
( x = (A ~ ) * b & ( for i, j being Nat st [i,j] in Indices x holds
x * i,j = ((Det A) " ) * (Det (ReplaceCol A,i,(Col b,j))) ) )
let A be Matrix of n,K; :: thesis: ( Det A <> 0. K implies for x, b being Matrix of K st len x = n & A * x = b holds
( x = (A ~ ) * b & ( for i, j being Nat st [i,j] in Indices x holds
x * i,j = ((Det A) " ) * (Det (ReplaceCol A,i,(Col b,j))) ) ) )
assume A1: Det A <> 0. K ; :: thesis: for x, b being Matrix of K st len x = n & A * x = b holds
( x = (A ~ ) * b & ( for i, j being Nat st [i,j] in Indices x holds
x * i,j = ((Det A) " ) * (Det (ReplaceCol A,i,(Col b,j))) ) )
A is invertible by A1, Th34;
then A ~ is_reverse_of A by MATRIX_6:def 4;
then A2: (A ~ ) * A = 1. K,n by MATRIX_6:def 2;
set MC = Matrix_of_Cofactor A;
set D = Det A;
A3: width (Matrix_of_Cofactor A) = n by MATRIX_1:25;
A4: len ((Matrix_of_Cofactor A) @ ) = n by MATRIX_1:25;
A5: width ((Matrix_of_Cofactor A) @ ) = n by MATRIX_1:25;
A6: width (A ~ ) = n by MATRIX_1:25;
A7: width A = n by MATRIX_1:25;
let x, b be Matrix of K; :: thesis: ( len x = n & A * x = b implies ( x = (A ~ ) * b & ( for i, j being Nat st [i,j] in Indices x holds
x * i,j = ((Det A) " ) * (Det (ReplaceCol A,i,(Col b,j))) ) ) )
assume that
A8: len x = n and
A9: A * x = b ; :: thesis: ( x = (A ~ ) * b & ( for i, j being Nat st [i,j] in Indices x holds
x * i,j = ((Det A) " ) * (Det (ReplaceCol A,i,(Col b,j))) ) )
A10: len A = n by MATRIX_1:25;
then A11: len b = n by A8, A9, A7, MATRIX_3:def 4;
x = (1. K,n) * x by A8, MATRIXR2:68;
hence A12: x = (A ~ ) * b by A8, A9, A6, A10, A7, A2, MATRIX_3:35; :: thesis: for i, j being Nat st [i,j] in Indices x holds
x * i,j = ((Det A) " ) * (Det (ReplaceCol A,i,(Col b,j)))
let i, j be Nat; :: thesis: ( [i,j] in Indices x implies x * i,j = ((Det A) " ) * (Det (ReplaceCol A,i,(Col b,j))) )
assume A13: [i,j] in Indices x ; :: thesis: x * i,j = ((Det A) " ) * (Det (ReplaceCol A,i,(Col b,j)))
A14: len (Col b,j) = n by A11, MATRIX_1:def 9;
Indices x = [:(Seg n),(Seg (width x)):] by A8, FINSEQ_1:def 3;
then A15: i in Seg n by A13, ZFMISC_1:106;
then A16: 1 <= i by FINSEQ_1:3;
A17: i <= n by A15, FINSEQ_1:3;
thus x * i,j = (Line (A ~ ),i) "*" (Col b,j) by A6, A12, A13, A11, MATRIX_3:def 4
.= (Line (((Det A) " ) * ((Matrix_of_Cofactor A) @ )),i) "*" (Col b,j) by A1, Th35
.= (((Det A) " ) * (Line ((Matrix_of_Cofactor A) @ ),i)) "*" (Col b,j) by A4, A16, A17, MATRIXR1:20
.= Sum (((Det A) " ) * (mlt (Line ((Matrix_of_Cofactor A) @ ),i),(Col b,j))) by A5, A11, FVSUM_1:82
.= ((Det A) " ) * ((Line ((Matrix_of_Cofactor A) @ ),i) "*" (Col b,j)) by FVSUM_1:92
.= ((Det A) " ) * ((Col (Matrix_of_Cofactor A),i) "*" (Col b,j)) by A3, A15, MATRIX_2:17
.= ((Det A) " ) * (Sum (LaplaceExpL (RLine (A @ ),i,(Col b,j)),i)) by A15, A14, Th31
.= ((Det A) " ) * (Det (RLine (A @ ),i,(Col b,j))) by A15, Th25
.= ((Det A) " ) * (Det ((RLine (A @ ),i,(Col b,j)) @ )) by MATRIXR2:43
.= ((Det A) " ) * (Det (RCol A,i,(Col b,j))) by Th19 ; :: thesis: verum