(a) Find three different permuted L U factorizations of the...

(a) Find three different permuted $L U$ factorizations of the matrix $A=\left(\begin{array}{rrr}0 & 1 & 2 \\ 1 & 0 & -1 \\ 1 & 1 & 3\end{array}\right)$.
(b) How many different permuted $L U$ factorizations does $A$ have?

Key Concepts

Gaussian Elimination with Pivoting

Gaussian elimination is an algorithm for solving systems of linear equations by transforming the matrix into an upper triangular form. Pivoting, which involves selecting and swapping rows (or columns) via permutation matrices, is used to avoid numerical instability and to ensure that the elimination process can proceed without encountering zero pivots. This strategy is the underpinning of permuted LU factorization and explains why multiple factorizations may exist based on different pivoting choices.

Permuted LU Factorization

This is an extension of the standard LU factorization where a permutation matrix (or matrices) is incorporated into the factorization process, typically written as PA = LU. Permuted LU factorization is critical when a matrix does not possess an LU factorization in its original order, necessitating row interchanges (captured by the permutation matrix) to obtain valid lower and upper triangular matrices.

Permutation Matrix

A permutation matrix is a square matrix that rearranges the rows (or columns) of another matrix when multiplied. It is formed by permuting the rows of an identity matrix. In the context of LU factorization, permutation matrices are used to rearrange the rows of the original matrix to prevent division by zero or to improve numerical stability during Gaussian elimination.

LU Factorization

This concept involves decomposing a matrix into a product of a lower triangular matrix (L) and an upper triangular matrix (U). It is fundamental in numerical linear algebra for solving systems of linear equations, inverting matrices, and computing determinants. The LU factorization is particularly useful because it reduces a matrix problem into simpler triangular system problems that are easier to solve.

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