The **Lu Decomposition Calculator** is used to factorize a square matrix with three rows and three columns into two matrices.

It decomposes a square matrix **A** into a **lower triangular** matrix** L** and an **upper triangular** matrix **U**.

The calculator takes a **square matrix** A with the **order** 3 x 3 as input and outputs the LU decomposition of the matrix that is the **product** of the matrices L and U. So, the matrix **A** can be written as:

**A = LU**

Where** L** and **U** are the lower triangular form and the upper triangular form of the **square matrix** **A** respectively. They both are special types of square matrices.

The **lower triangular** matrix is specified by having all the entries equal to zero which are **above** the main diagonal. Similarly, the **upper triangular** matrix has all the elements **below** its main diagonal equal to zero.

In **LU decomposition**, the entries above the main diagonal in the lower triangular matrix and the entries below the main diagonal in the upper triangular matrix are **not altered**.

The calculator only **changes** the remaining entries according to the matrix A.

The user can use this calculator to solve a system of **three linear equations** using **LU decomposition**. The coefficients in the system of three linear equations can be written in matrix form as:

**AX = B**

Where **X** is the **unknown** matrix. In LU decomposition, the matrix **A** is replaced with the product of matrices **LU** as follows:

**LUX = B**

The matrices **L** and **U** will be obtained by using this calculator. If we suppose UX=Y and substitute in the above equation, it gives:

**LY = B**

First solving for **Y** in the above equation and then putting the values of Y in UX = Y and then solving for **X** gives the solution of the system of three linear equations using **LU** decomposition.

## What Is a LU Decomposition Calculator?

**The Lu Decomposition Calculator is an online tool that is used to decompose a 3 x 3 square matrix A into the product of an upper triangular 3 x 3 square matrix U and a lower triangular 3 x 3 square matrix L.**

## How To Use the Lu Decomposition Calculator

The user can use the Lu Decomposition Calculator by following the steps given below:

### Step 1

The user must first enter the **first row** of the 3 x 3 square matrix A in the calculator’s input window. The three elements should be entered in curly brackets with commas separating them in the block labeled, “**Row 1**”.

For the **default** example, the elements of the first row entered are { 3,1,6 }.

### Step 2

The user must now enter the **second row** of the matrix A in the input tab of the calculator.

To form a square matrix, the user must enter three entries in the block labeled, “**Row 2**” in between flower brackets with commas separating the elements.

The user enters the second row as { -6,0,-16 } for the **default** example.

### Step 3

The **third row** of the square matrix A should be entered in the block titled, “**Row 3**” in the calculator’s input window. For the **default** example, the entries of the third row are { 0,8,-17 }.

### Step 4

The user must now press the “**Submit**” button for the calculator to process the input 3 x 3 matrix entered by the user.

### Output

The calculator displays the output in the following **two windows** by computing the LU decomposition of the input matrix.

#### Input

The calculator **interprets the input** and displays the three input rows in the form of a 3 x 3 square matrix in this output window.

For the **default** example, the calculator shows the Input interpretation as follows:

\[ LU \ decomposition = \begin{bmatrix} 3 & 1 & 6 \\ -6 & 0 & -16 \\ 0 & 8 & -17 \\ \end{bmatrix} \]

#### Result

The calculator computes the **LU decomposition** of the square matrix **A** by using the equation:

**A = LU**

For the **default** example, the calculator displays the **A**,** L**, and **U** as follows:

\[ A = \begin{bmatrix} 3 & 1 & 6 \\ -6 & 0 & -16 \\ 0 & 8 & -17 \\ \end{bmatrix} \]

\[ L = \begin{bmatrix} 1 & 0 & 0 \\ -2 & 1 & 0 \\ 0 & 4 & 1 \\ \end{bmatrix} \]

\[ U = \begin{bmatrix} 3 & 1 & 6 \\ 0 & 2 & -4 \\ 0 & 0 & -1 \\ \end{bmatrix} \]

## Solved Example

The following example is solved through the Lu Decomposition Calculator.

### Example 1

For the square matrix **A** given as:

\[ A = \begin{bmatrix} 1 & 1 & 1 \\ 4 & 3 & -1 \\ 3 & 5 & 3 \\ \end{bmatrix} \]

Compute the matrices **L** and **U** from the **LU decomposition** method.

### Solution

The user must enter the **three rows** as { 1,1,1 } , { 4,3, -1 } and { 3,5,3 } in the three input blocks of the calculator.

After submitting the three input rows, the calculator displays the 3 x 3 **Input** square matrix as follows:

\[ LU \ decomposition = \begin{bmatrix} 1 & 1 & 1 \\ 4 & 3 & -1 \\ 3 & 5 & 3 \\ \end{bmatrix} \]

The calculator computes the **LU decomposition** of the input matrix A and displays the three matrices as follows:

\[ A = \begin{bmatrix} 1 & 1 & 1 \\ 4 & 3 & -1 \\ 3 & 5 & 3 \\ \end{bmatrix} \]

\[ L = \begin{bmatrix} 1 & 0 & 0 \\ 4 & 1 & 0 \\ 3 & -2 & 1 \\ \end{bmatrix} \]

\[ U = \begin{bmatrix} 1 & 1 & 1 \\ 0 & -1 & -5 \\ 0 & 0 & -10 \\ \end{bmatrix} \]