Finding the eigenvalues with an augmented matrix 2x2

Hi, I somehow have a problem with obtaining the row reduction in an augumented matrix to find lamda. Currently I have this code:

% A
I = eye(2)
m1 = [7 3; 3 -1]
Lamda = 8                   % The value for λ
% B
LaI = I * Lamda            % Identity matrix - value of λ
m2s = m1 - LaI            % Caluculating the new matrix
% C
zc = zeros(size(m2s,1),1)     % creating a zero column
m3s = [m2s, zc]                     % adding the zero column at the end
% D
A3 = [-1 3;
    3 -9]
b = [0; 0]
[L, U, P] = lu(A3)             % L = all multipliers, U = upper triangular matrix,  P = row interchanges
% factors to solve linear system
y = L\(P*b) % Forward substitution
x = U\y %Backward substitution

In part D there I get this warning:

Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 5.551115e-17. <warning </warning>

I got 2 question:

First, how do I fix it?

Second how can I define the relationship between x1 and x2?

Help is much appreciated <3 Thanks!

[David Goodmanson]

Hi Jonas,

in %D, the matrix A3 is singular. So solving the Ax = b problem by LU decomposition or any other method is not going to work.

A3 is the same as m2s, which you obtained from m1 - 8*I, where 8 is one of the eigenvalues of m1. Therefore m2s is singular by construction. It looks like you need to go bakc and reassess the entire process.