Page 759 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

10.

Let
11.

(a) Find an -decomposition of A.

(b) Express A in the form , where is lower triangular with 1's along the main diagonal, is upper

triangular, and D is a diagonal matrix.

(c) Express A in the form , where is lower triangular with 1's along the main diagonal and is upper
triangular.

12.
(a) Show that the matrix

has no -decomposition.
(b) Find a -decomposition of this matrix.

Let
13.

(a) Prove: If , then A has a unique -decomposition with 1's along the main diagonal of L.

(b) Find the -decomposition described in part (a).

Let be a linear system of n equations in n unknowns, and assume that A is an invertible matrix that can be
14. reduced to row-echelon form without row interchanges. How many additions and multiplications are required to solve

the system by the method of Example 1?
Note Count subtractions as additions and divisions as multiplications.
Recall from Theorem 1.7.1b that a product of lower triangular matrices is lower triangular. Use this fact to prove that the
15. matrix L in 8 is lower triangular.