Last Updated on July 30, 2026 by Maged kamel
How to derive the expression for L-U values for a -3×3 matrix?
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This explains the terms for both the upper and lower L-U values in Doolittle’s method for the 3×3 matrix.
The matrix A of 3×3 is represented by(aij) where i is the row number, and j is the column number.
The matrix A (3×3) can be represented as the product of the Lower triangular matrix L (3×3) and the upper triangular matrix U (3×3).
For the terms of the L matrix, we also use the notation aij. For that L matrix, the diagonal is 1, and we have three diagonals: a11 = a22 = a33 = 1.
For a12, the element in the first row and second column =a13=0. Below the diagonal, all remaining elements are nonzeros.
On the contrary, the upper triangular matrix has U11, U12, and U13, and all the diagonals are non-zero, while all elements below the diagonal represented by U11, U22, and U33 are all zeros.
Step-1: Equating the product of L-U values for a 3×3 matrix with the value of matrix A.
Since matrix A is the product of LU matrices, we can equate the value of the elements of A with the product of LU for the corresponding items.
The product of L*U is represented by a 3×3 matrix, as we can see in the next slide image. The first row contains (U11, U12, and U13), which are equal to the first row of matrix A; hence, we can proceed to the next step.
Step-2: Get the expression for L-U elements for the upper and lower matrix for a 3 x 3 matrix.
1-Equate the first element of (L*U) decomposition to the first element a11 of the matrix A as follows: U11=a11. Similarly, we can equate the second and the third elements to the corresponding elements of matrix A.
So, U12 = a12 and U13=a13.
2-We will move to the second row of the L*U matrix to get the expression for L-U values.
3- We equate L21*U11 to a21, also, L21*U12+U22 to a22, and L21*U13+U23 to a23, we will use the derived values for U11 as a11, U12 as a12 from which we have estimated from step 1.
4- We can get the values of L21 and L32.
We set L21*U12 + U22 equal to a22; we substitute u12 = a12 and L21 = a21/a11.
5- We can get U22, and U23 as shown in the next slide image.
The value of L32 is shown in more detail on the following slide.
To get the value of U33, we set L31*U13 + L32*U23 + U33 equal to a33, and use the estimated values to obtain an expression for U33. Please refer to the following slide image.
The following slide shows the component of the lower matrix L that we need to create the U1 matrix to find the value of L32.
We need to create the U1 matrix; we multiply the second row of matrix A by -a21/a11*R1 and add it to the second row. We multiply the third row of matrix A by -a31/a11*R1 and add it to the third row. In that matrix, U21 = U31 = 0.
The element u22=a22-a21/a11*a12, this is the pivot element we need to get the value of L32.
The next slide image shows how to get the value of L32. It is equal to ((a32-(a31/a11)*a12/a22-a21*(a12/a11). We take U21 as a pivot.
The following slide shows how to obtain the value of U33.
Thus, we have come to the end of the method to derive the expression for L-U values for an A 3×3 matrix.
In the next post, we will use a solved problem to apply this method. A discussion on whether all matrices will have LU decomposition or not. You can download the PDF file for this post
This is a direct link to the next post, which is the First solved problem-LU decomposition for the 3×3 matrix.
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This is the Omni calculator for estimating various items of linear algebra -LU Decomposition Calculator.
This is a link to another –Calculator for matrices.