**True or False Statements: LU Factorization****Instructions:** Determine whether the following statements are true or false. Justification is required for each answer.**(a)** In finding an LU factorization, only row replacement is allowed, where $ R_i $ is replaced by $ R_i + kR_j $ for $ i > j $.**(b)** Every matrix has an LU factorization.

Answered: Mark the following either true or…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

**True or False Statements: LU Factorization**

**Instructions:** Determine whether the following statements are true or false. Justification is required for each answer.

**(a)** In finding an LU factorization, only row replacement is allowed, where $ R_i $ is replaced by $ R_i + kR_j $ for $ i > j $.

**(b)** Every matrix has an LU factorization.

Expert Solution

Step 1: Justification of part a

False,

Row replacement is not the only operation allowed in LU factorization. LU factorization, specifically using Gaussian elimination, involves three types of elementary row operations:

Row interchange: Interchanging the positions of two rows.
Row scaling: Multiplying a row by a nonzero constant.
Row addition: Adding a multiple of one row to another row.

These operations are used to transform the original matrix into an upper triangular form $left parenthesis U right parenthesis$ while simultaneously constructing a lower triangular matrix $left parenthesis L right parenthesis$ that keeps track of the row operations performed. The row operations are chosen in a way that the resulting matrix $U$ is upper triangular, and the matrix $L$ is lower triangular with ones on the diagonal.

So, row scaling (multiplying rows by constants) and row addition (adding multiples of rows) are indeed allowed operations in LU decomposition.