################ (2021) Dr.W.G. Lindner, Leichlingen DE
###       craBox        Functions for Cramer rule etc.
################

Replace(p,M,i) = do(X=M, 
                    Xt=transpose(X), 
                    Xt[i]=p, 
                    transpose(Xt))

-- 2 x 2
Cramer2(A,B) = ( det(Replace(B,A,1))/det(A),
                 det(Replace(B,A,2))/det(A) )

-- 3 x 3
Cramer3(A,B) = ( det(Replace(B,A,1))/det(A),
                 det(Replace(B,A,2))/det(A),
                 det(Replace(B,A,3))/det(A) )

-- n x n 
Cramer(A,B) = do( n = dim(A,1),
                  Z = zero(2,n),       
                  Y = Z[1],   
                  for( i,1,n, 
                       Y[i] = det(Replace(B,A,i))/det(A) ),
                  Y )

CramerAdj(A,B) = dot(adj(A),B)/det(A)

Cramer3x(M,R)= do( M=transpose(M),         
                   Box(   R, M[2], M[3]) /  
                   Box(M[1], M[2], M[3]))  


-- MinorMatrices
MiMa(A)= do( e = quote(e), i = quote(i),    
             MM = zero(3,3,2,2),            
             for(k,1,3,
             for(j,1,3,
                 MM[k,j] = minormatrix(A,k,j))), 
             MM)  

-- CofactorsMatrix
CoMa(A) = do( e = quote(e), i = quote(i), 
              MM = zero(3,3),           
              for(k,1,3,
              for(j,1,3,
                 MM[k,j] = cofactor(A,k,j))),
              MM)             

detLaplace(A,i)  = sum(j,1,dim(A,1),  (-1)^(i+j)*A[i,j]*minor(A,i,j) )

detCofactor(A,i) = sum(j,1,dim(A,1),  A[i,j]*cofactor(A,i,j) )

invAdj(M) = adj(M)/det(M)

DET3(M) = dot(M[1],cross(M[2],M[3]))


#################  craBox END ####################