206 CHAPTER 10. A FEW FACTORIZATIONS

and so you can now tell what the various quantities equal. From the first column, youneed a = 1,x = 1,y = 2. Now go to the second column. You need d = 2,xd + b = 3 sob = 1,yd + zb = 3 so z = −1. From the third column, h = 0,e = 2,c = 6. Now from thefourth column, j = 2, i =−1, f =−5. Therefore, an LU factorization is 1 0 0

1 1 02 −1 1

 1 2 0 2

0 1 2 −10 0 6 −5

 .

You can check whether you got it right by simply multiplying these two.

10.3 Using Multipliers To Find An LU FactorizationThere is also a convenient procedure for finding an LU factorization. It turns out that itis only necessary to keep track of the multipliers which are used to row reduce to uppertriangular form. This procedure is described in the following examples.

Example 10.3.1 Find an LU factorization for A =

 1 2 32 1 −41 5 2

Write the matrix next to the identity matrix as shown. 1 0 0

0 1 00 0 1

 1 2 3

2 1 −41 5 2

 .

The process involves doing row operations to the matrix on the right while simultaneouslyupdating successive columns of the matrix on the left. First take −2 times the first row andadd to the second in the matrix on the right. 1 0 0

2 1 00 0 1

 1 2 3

0 −3 −101 5 2

Note the way we updated the matrix on the left. We put a 2 in the second entry of the firstcolumn because we used −2 times the first row added to the second row. Now replace thethird row in the matrix on the right by −1 times the first row added to the third. Noticethat the product of the two matrices is unchanged and equals the original matrix. This isbecause a row operation was done on the original matrix to get the matrix on the right andthen on the left, it was multiplied by an elementary matrix which “undid” the row operationwhich was done.

The next step is  1 0 02 1 01 0 1

 1 2 3

0 −3 −100 3 −1

