Активируйте этот код перед использованием любых ячеек Linked Sage Cells
xxxxxxxxxxM2=matrix(SR,2,var('a,b,c,d'))M3=matrix(SR,3,var('a,b,c,d,e,f,g,h,i'))M4=matrix(SR,4,var('a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p'))A=random_matrix(ZZ,3,x=0,y=100)B=random_matrix(ZZ,3,x=-100,y=0) C=random_matrix(ZZ,4,x=-100,y=100)xxxxxxxxxxfor M in [M2,M3,A,B,C]: pretty_print((M,M.det()))Активируйте этот код перед использованием любых ячеек Linked Python Cells
xxxxxxxxxximport numpy,sympysympy.init_printing(use_unicode=True)A1=numpy.matrix( numpy.random.randint(0,100,size=(3,3)))B1=numpy.matrix( numpy.random.randint(-100,0,size=(3,3)))C1=numpy.matrix( numpy.random.randint(-100,100,size=(4,4)))A2=sympy.Matrix([[-3,-3,1],[-4,-7,7],[2,2,5]])B2=sympy.Matrix([[2,10,7],[-5,-6,-2],[-8,4,-8]])C2=sympy.Matrix([[-8,-4,5,2],[3,-5,8,-2], [5,4,-8,1],[7,-2,9,5]])mm=sympy.symbols(' '.join(list('m_%d%d' %(i,j) for i in [1,2] for j in [1,2])))M=sympy.Matrix(int(2),int(2),mm) nn=sympy.symbols(' '.join(list('n_%d%d' %(i,j) for i in [1,2,3] for j in [1,2,3])))N=sympy.Matrix(int(3),int(3),nn)xxxxxxxxxxfor m in [A1,B1,C1]: print([m,round(numpy.linalg.det(m))])for m in [A2,B2,C2]: print([m,m.det()])for m in [M,N]: sympy.pprint(m) sympy.pprint(m.det())Активируйте этот код перед использованием любых ячеек Linked R Cells
xxxxxxxxxx%%rA<-matrix(sample.int(100,size=3*3,replace=TRUE), nrow=3,ncol=3)B<-matrix(sample.int(100,size=3*3,replace=TRUE), nrow=3,ncol=3)C<-matrix(sample.int(100,size=4*4,replace=TRUE), nrow=4,ncol=4)library(Matrix)xxxxxxxxxx%%rc(det(A),det(B),det(C))xxxxxxxxxxpretty_print('A.det()=A.T.det(): '+\ str(A.det()==A.T.det()))pretty_print('(A*B).det()=(B*A).det()=A.det()*B.det(): '+\ str((A*B).det()==(B*A).det()==A.det()*B.det()))pretty_print('(-2*A).det()=(-2)^A.dimensions()[0]*A.det(): '+\ str((-2*A).det()==(-2)^A.dimensions()[0]*A.det()))# linear dependence[matrix([[1,2,3],[0,0,0],[7,8,9]]).det(), matrix([[1,2,3],[2,4,6],[7,8,9]]).det()]xxxxxxxxxxprint(round(numpy.linalg.det(A1))==\round(numpy.linalg.det(A1.T)),\round(numpy.linalg.det(A1*B1))==\round(numpy.linalg.det(B1*A1))==\round(numpy.linalg.det(A1))*round(numpy.linalg.det(B1)),\round(numpy.linalg.det(-2*A1))==\-2**A2.shape[0]*round(numpy.linalg.det(A1))) # linear dependenceround(numpy.linalg.det(numpy.matrix([[1,2,3],[0,0,0],[7,8,9]]))),\round(numpy.linalg.det(numpy.matrix([[1,2,3],[2,4,6],[7,8,9]])))xxxxxxxxxxprint([A2.det()==A2.T.det(), (A2*B2).det()==(B2*A2).det()==A2.det()*B2.det(), (-2*A2).det()==(-2)^A2.shape[0]*A2.det()])# linear dependence[sympy.Matrix([[1,2,3],[0,0,0],[7,8,9]]).det(),sympy.Matrix([[1,2,3],[2,4,6],[7,8,9]]).det()]xxxxxxxxxx%%rprint(c(all.equal(det(A),det(t(A))), all.equal(det(A%*%B), det(B%*%A),det(A)*det(B)), all.equal(det(-2*A),(-2)^(dim(A)[1])*det(A))))# linear dependencec(det(matrix(c(1,3,5,0,0,0,2,4,6), nrow=3,ncol=3)), det(matrix(c(1,3,5,2,6,10,2,4,6), nrow=3,ncol=3)))xxxxxxxxxx%%html$SIGN_{(3,3)} = \begin{pmatrix} + & - & + \\ - & + & - \\ + & - & + \\ \end{pmatrix} \ \SIGN_{(4,4)} = \begin{pmatrix} + & - & + & - \\ - & + & - & + \\ + & - & + & - \\ - & + & - & + \end{pmatrix}$xxxxxxxxxx[M3.minors(2),M4.minors(3), A.minors(2),B.minors(2),C.minors(3)]xxxxxxxxxx# разложение определителя по первому столбцуM3.det()==M3[0,0]*M3.minors(2)[8]-\M3[1,0]*M3.minors(2)[5]+M3[2,0]*M3.minors(2)[2]xxxxxxxxxxdef minor(m,i,j): return [row[:j]+row[j+1:] for row in (m[:i]+m[i+1:])]def deternminant(m): if len(m)==2: return m[0][0]*m[1][1]-m[0][1]*m[1][0] det=0 for c in range(len(m)): det+=((-1)**c)*m[0][c]*deternminant(minor(m,0,c)) return det[minor([[1,2,3],[-4,5,-6],[-7,8,-9]],1,1),deternminant([[1,2,3],[-4,5,-6],[-7,8,-9]])]xxxxxxxxxxsympy.pprint(A2)sympy.pprint(A2.minorMatrix(0,0))xxxxxxxxxx# разложение определителя по первому столбцуB2.det()==B2[0,0]*B2.minorMatrix(0,0).det()-\B2[1,0]*B2.minorMatrix(1,0).det()+\B2[2,0]*B2.minorMatrix(2,0).det()xxxxxxxxxx%%rminor_matrix<-function(A,i,j) A[-i,-j]minor<-function(A,i,j) det(A[-i,-j])cofactor<-function(A,i,j) (-1)^(i+j)*minor(A,i,j)c(minor_matrix(A,1,2),minor(A,1,2),cofactor(A,1,2))xxxxxxxxxx%%rc(det(A), (A[1,1]*minor(A,1,1)- A[2,1]*minor(A,2,1)+ A[3,1]*minor(A,3,1)))xxxxxxxxxxxxxxxxxxxximport sympy,numpy,pylabxxxxxxxxxx%%rxxxxxxxxxx