Last Updated on July 30, 2026 by Maged kamel
How to use the LU decomposition method for a 2×2 matrix?
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LU decomposition method for the 2×2 matrix?
Now we will start with the lower- and upper-triangular components of the LU decomposition. There are two methods to find the lower and upper matrices: Doolittle’s method and Crout’s method. In this post, we will introduce Doolittle’s method for LU decomposition. The next slide image summarises the content of this post.
The lower-triangular matrix property holds for a matrix. It has a nonzero diagonal, and the lower corner has a nonzero value. For the upper corner, each element will be=0.
On the contrary, the upper matrix has a zero diagonal, with the upper corner nonzero and the lower corner zero. Just why is this called a lower matrix? It is enclosed by the diagonal and the lower bottom, while the upper matrix is enclosed by the diagonal and the upper right-hand side.
For the symbols. Thus, the matrix can be decomposed using the LU decomposition method.
LU decomposition method by using an example of a 2×2 matrix.
For the symbols, we will write an A matrix (2 3 3 4), which means a11, a12, a21, and a22. We have six unknowns: L11, L21,l22 ,U11, U12, and U22.
Due to the multiplication of the lower by the upper Matrix, we have only four equations.
According to the LU triangulation method, the two elements L11 and L22 will be equal to 1. Hence, we can solve the four equations to get the values of the remaining unknowns.
Since A = the lower multiplication by the upper matrices, we will perform the multiplication of the lower and upper matrices and equate the product to the elements of matrix A.
For example, if we multiply the first row by the First Column, we will get the value of 2, which we express as (1*U11+0)=2.
The product of the First row and the second column is 3. Meanwhile, a22, equal to 4, will be equivalent to the multiplication of the second row by the second column, L21 *U12)+(1*U22)=4. From equation I, we have U11 = 2.
From equation II, we have U12 = 3. Substitute the value of U11 in equation three to get the value of L21, which is equal to 3/2.
Finally, we can get the value of U22 from equation four after substituting the values of L21 and U12. U22 will be equal to -1/2. Remaining for us the minus(-) (1/2) is denoted by U22, which is = 4 – 3, which is L21, 2 * 3 = -1/2. or U22=a22-L21*U12.
If we put the Matrix on the left-hand side and the corresponding values after solving the four equations, we will find an interesting observation: a11 will go to U11, and a12 will go to U12, as if we are taking these values from here to there.
Consider a11 as a pivot; divide a21/a11, which is equal to 3/2, to come to a value of L21. This is from the equations we have solved.
While multiplying a21/a11 by minus will be multiplied by L21 and deducted from a22 to get u22. Please refer to the next slide image.
Do all 2×2 matrices have LU decomposition?
Not all 2×2 matrices have an LU decomposition. In the case of a determinant value, LU decomposition can not be considered because it will lead to a U matrix with a zero value on the diagonal.
To estimate the values of the elements of the upper Matrix, we have U11 = a11 = 1 and U12 = a22 = 5 by definition. U12=0. Finally, U22=a12-L21*u12=10*(2*5)=0. U22 cannot equal zero, since U is an upper matrix with non-zero diagonal elements. This is the upper matrix for matrix B.
The expression of B=LU is not acceptable because a U matrix cannot have U22=0.
This is the PDF file for the illustration for Post 2.
The following post, post 3, is the Best illustration of how to solve for x-y for two equations using L/U decomposition.
HELM-Helping Engineers Learn Mathematics.
This is the Omni calculator for estimating various items of linear algebra -LU Decomposition Calculator.
This is a link to another –Calculator for matrices.