Wrong answers given by MATLAB LU Factorization

LU factorization or decomposition method is supposed to decompose a square matrix to a lower matrix and a upper matrix. However, I tried to test some 2 by 2 matrix, the MATLAB LU method sometimes give me 2 upper matrix. Such an original matrix as:

A=[4 , 3 ; 6 , 3]

The answer given by MATLAB:

[L,U]=lu(A)
L=[0.667,1 ; 1, 0]
U=[6, 3 ; 0, 1]

The correct answer should be

L=[1 , 0 ; 1.5, 1]
U=[4 , 3 ; 0, -1.5]

I don't understand why MATLAB gives me such answer

Please, someone help me!

Answer 1

[Sean de Wolski]

The behavior you are seeing is described in Example 1 of the lu documentation:

http://www.mathworks.com/help/releases/R2013b/matlab/ref/lu.html

[Zhuo]

Hello, Thank you very much for your help.

I think the only way to let MATLAB to give me the correct format of lower matrix is to use: [L,U,P]=lu(A) But I have to pre-multiply the inverse of P in order to convert back to A Do you know there is a way to let MATLAB to give me an upper and a lower matrix straight away without involving intermediate matrix, such as P ?

Answer 2

[José-Luis]

Both answers are valid.

 A = [2/3,1 ; 1, 0] *[6, 3 ; 0, 1]  
 B = [1 , 0 ; 1.5, 1] * [4 , 3 ; 0, -1.5]
 A == B

[Zhuo]

Thank you very much! I know both answers are valid. However, it is very necessary to write one of the matrix in lower matrix format which is easier for my program to take further calculations.

[José-Luis]

I don't understand what you mean.