Question

Find P, L, and U such that PA=LU (i.e., the LU decomposition with row exchanges), where:

A) P is the permutation matrix, L is the lower triangular matrix, and U is the upper triangular matrix
B) P is the upper triangular matrix, L is the lower triangular matrix, and U is the permutation matrix
C) P is the identity matrix, L is the lower triangular matrix, and U is the upper triangular matrix
D) None of the above

Answer 1

In the LU decomposition with row exchanges, P is the permutation matrix, L is the lower triangular matrix, and U is the upper triangular matrix, making option (A) the correct answer.

Step-by-step explanation:

The question is asking to find matrices P, L, and U such that PA = LU, where A is a given matrix and the decomposition includes row exchanges if necessary. This type of factorization is known as the LU decomposition with partial pivoting.

The correct answer is (A):

1. P is the permutation matrix: This matrix is used to record the row exchanges that are necessary for the factorization.
2. L is the lower triangular matrix: This matrix has elements below the main diagonal and contains the multipliers from the Gaussian elimination process.
3. U is the upper triangular matrix: This matrix has elements above the main diagonal and results from the Gaussian elimination.

Options B and C are incorrect because in option B U and P are misplaced and in option C there is no permutation matrix to account for row exchanges which is a necessary component of LU decomposition with partial pivoting.