[SOLVED] MATH307 Individual Homework21

Instructions: Read textbook pages 114 to 117 before working on the homework problems. Show all steps to get full credits.

1. Let A be a n×n matrix with a₁₁ 6=0, the first step in LU decomposition is to introduce zeros below the first diagonal a₁₁. This can be done by multiplying A by a lower triangular matrix L₁ that is equal to the n × n diagonal matrix

except the first column looks like `₁ =              −l₃₁                with

…

−`n1

2,3,···n. It is obvious that. Prove. This is the first stroke of luck in LU decomposition: find the inverse of L₁ can be done by simply negativing the entries below the first diagonal.

^Ñ 1 2 1 3^é

2. Find the general solution to Ax = b with A = −3 2 1 0 and b =

3 2 1 1

Ñ 2 é

−5 . You may use some of the information from the previous problem.

2

^Ñ−1 2 1 0 2 ^{é Ñ}−1^é

3. Given the matrix A = 2 0 0 3 −1 ,b =       0         ,c ∈ R,

−1 6 3 3 5                              c

• For which value of c does the equation Ax = b have a solution?
• After choosing c so that the system has a solution, find a particular solution to Ax = b.