According to my textbook "Matrix Operations for
Engineers and Scientists - An Essential Guide in Linear Algebra" by the late Alan Jeffrey the following system of equations
is impossible. To quote the author: *System (a) has no solution. This can be shown in more than one way. The most elementary way being to solve the first three equations for , and , and then to substitute these values into the last equation to show that they do not satisfy it. Thus the last equation contradicts the other three, so there can be no solution set.*
Nevertheless, according to my understanding of the Kronecker-Capelli theorem the system under question has a unique solution.
The rank of the matrix of the coefficients of unknowns is 3. The rank of the augmented matrix of the system is also 3. Finally the number of unknowns is 3 as well.
The reduced row echelon form of the matrix that I found is
Thus, according to my understanding
The following script Matlab verifies my findings.
A = [1 -2 2; 1 1 -1; 1 3 -3; 1 1 1]; b = [6; 0; -4; 3];
if rank(A) == rank([A b])
size_A = size(A);
if rank(A) == size_A (2)
disp('There is a unique solution, which is:')
x = A\b
disp('There is an infinite number of solutions')
disp('The augmented matrix of the reduced system is:')
disp('There are no solutions.')
There is a unique solution, which is:
What am I missing here?