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_0:24;
A4: len ((Matrix_of_Cofactor A) @) = n by MATRIX_0:24;
A5: width ((Matrix_of_Cofactor A) @) = n by MATRIX_0:24;
A6: width (A ~) = n by MATRIX_0:24;
A7: width A = n by MATRIX_0:24;
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_0:24;
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_3:33; :: 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_0:def 8;
Indices x = [:(Seg n),(Seg (width x)):] by A8, FINSEQ_1:def 3;
then A15: i in Seg n by A13, ZFMISC_1:87;
then A16: 1 <= i by FINSEQ_1:1;
A17: i <= n by A15, FINSEQ_1:1;
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_1:68
.= ((Det A) ") * ((Line (((Matrix_of_Cofactor A) @),i)) "*" (Col (b,j))) by FVSUM_1:73
.= ((Det A) ") * ((Col ((Matrix_of_Cofactor A),i)) "*" (Col (b,j))) by A3, A15, MATRIX_0:59
.= ((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