Enter the original matrix into the calculator to calculate the LU decomposition, where A is the original matrix, L is the lower triangular matrix, and U is the upper triangular matrix.

## Lu Decomposition Formula

The following formula is used to calculate the LU decomposition of a matrix:

A = LU

Where:

- A is the original matrix
- L is the lower triangular matrix
- U is the upper triangular matrix

The LU decomposition is a factorization of a matrix into the product of a lower triangular matrix and an upper triangular matrix. It is commonly used in numerical analysis and linear algebra to solve systems of linear equations and calculate determinants.

## What is a Lu Decomposition?

LU Decomposition, also known as LU factorization, is a method in numerical analysis for solving linear equations, inverting matrices, and computing determinants. The term “LU” stands for “Lower Upper”, and it decomposes a matrix as the product of a lower triangular matrix and an upper triangular matrix. The lower triangular matrix has ones on the diagonal and the elements above the diagonal are zero, while the upper triangular matrix has zeros below the diagonal. This decomposition is mainly used to simplify the solution of a system of linear equations, as it can be applied to the system’s matrix. LU Decomposition is particularly useful when the system needs to be solved repeatedly for different right-hand sides, as it reduces the computational cost.

## How to Calculate Lu Decomposition?

The following steps outline how to calculate the LU Decomposition:

- First, write the given matrix A.
- Next, perform Gaussian elimination on matrix A to obtain an upper triangular matrix U.
- Next, record the elementary row operations used during Gaussian elimination to obtain a lower triangular matrix L.
- Finally, write the LU Decomposition as A = LU.

**Example Problem:**

Use the following matrix as an example problem to test your knowledge:

A = [[2, 4, 6], [1, 3, 5], [7, 8, 9]]