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LU Decomposition

#### Part 4: A Modified Decomposition

In this part, we will examine matrices for which our methods for finding an **LU** decomposition fail. We will modify the **LU** decomposition algorithm so it works for *any* matrix.

- Enter the matrix
**R** in your worksheet. As you did in Part 1, multiply **R** by elementary matrices to reduce it to a matrix **U** in row echelon form. (Note that a row swap is necessary!) Find a matrix **P** such that **PR = U**. Is **P** unit lower triangular? Explain why or why not.
- Find a matrix
**L** such that **R = LU**. Have you found an **LU** decomposition for **R**? Explain why or why not.
- Ask your computer algebra system to find an
**LU** decomposition of **R**. Is the product of the reported matrices **L** and **U** the same as **R**? If not, explain how **R** and the product **LU** differ. How does this relate to your answers to the previous questions?
- Some texts define the
**LU** decomposition for any matrix by allowing **L** to be *permuted* lower triangular, meaning that some permutation of the rows of **L** can turn **L** into a unit lower triangular matrix. Determine whether or not your helper application found such a modified **LU** decomposition for **R**.
- Enter the matrix
**S** in your worksheet, and use the commands there to reduce **S** to row echelon form. Use what you have learned to find a modified **LU** decomposition for **S** (without using a built-in helper application command). Carefully explain how you found the permuted lower triangular matrix **L**. Check your answer using the command(s) available in your helper application.

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