208 CHAPTER 10. A FEW FACTORIZATIONS

10.4 Solving Systems Using An LU FactorizationOne reason people care about the LU factorization is it allows the quick solution of systemsof equations. Here is an example.

Example 10.4.1 Suppose you want to find the solutions to

 1 2 3 24 3 1 11 2 3 0



xyzw

=

 123

 .

Of course one way is to write the augmented matrix and grind away. However, thisinvolves more row operations than the computation of the LU factorization and it turns outthat the LU factorization can give the solution quickly. Here is how. The following is anLU factorization for the matrix. 1 2 3 2

4 3 1 11 2 3 0

=

 1 0 04 1 01 0 1

 1 2 3 2

0 −5 −11 −70 0 0 −2

 .

Let Ux = y and consider Ly = b where in this case, b =(1,2,3)T . Thus 1 0 04 1 01 0 1

 y1

y2

y3

=

 123

which yields very quickly that y =

(1 −2 2

)T. Now you can find x by solving

Ux = y. Thus in this case,

 1 2 3 20 −5 −11 −70 0 0 −2



xyzw

=

 1−22

which yields

x =(

75 t− 3

595 −

115 t t −1

)T, t ∈ R.

10.5 Justification For The Multiplier MethodWhy does the multiplier method work for finding the LU factorization? Suppose A is amatrix which has the property that the row reduced echelon form for A may be achievedusing only the row operations which involve replacing a row with itself added to a multipleof another row. It is not ever necessary to switch rows. Thus every row which is replacedusing this row operation in obtaining the echelon form may be modified by using a rowwhich is above it.