# Solving an overdetermined system of linear equations

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Dimitrios Anagnostou on 6 May 2021
Answered: James Tursa on 6 May 2021
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.
% Script file lineq.m
% Solves the set Ax = b, given A and b.
clc; clear;
A = [1 -2 2; 1 1 -1; 1 3 -3; 1 1 1]; b = [6; 0; -4; 3];
% Check the ranks of A and [A b].
if rank(A) == rank([A b])
% The ranks are equal.
size_A = size(A);
% Does the rank of A equal the number of unknowns?
if rank(A) == size_A (2)
% Yes. Rank of A equals the number of unknowns.
disp('There is a unique solution, which is:')
x = A\b % Solve using left division.
else
% Rank of A does not equal the number of unknowns.
disp('There is an infinite number of solutions')
disp('The augmented matrix of the reduced system is:')
rref([A b]) % Compute the augmented matrix.
end
else
% The ranks of A and [A b] are not equal.
disp('There are no solutions.')
end
with output
There is a unique solution, which is:
x =
2.0000
-0.5000
1.5000
What am I missing here?
: https://i.stack.imgur.com/dbtY3.png

James Tursa on 6 May 2021
Since you can simply plug in those values to verify that they satisfy all four equations, the conclusion would be that you have uncovered a typo in the book.

R2020b

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