Given A =(1,3,1),(2,7,4),(3,8,5), find elementary matrices E 1, E 2, E 3 such th
ID: 3114403 • Letter: G
Question
Given
A=(1,3,1),(2,7,4),(3,8,5),
find elementary matrices E1, E2, E3 such that
E3E2E1A=U,
where U is an upper triangular matrix. Make your first step E1 puts a 0 in the (3,1) position. (Notice that A can be put in upper triangular form by three elementary row operations. Find elementary matrices corresponding to each of them. Remember order is important!)
E1=
E2=
E3=
(All blanks must be filled in to receive any credit on this one.)
Now notice that A=LU, where L=(E3E2E1)^1.
Compute L= .
Notice that L is lower triangular with ones on the diagonal. This is called the LU factorization of A.
Explanation / Answer
We presume that the given vectors are row vectors. Then A is a 3x3 matrix
1
3
1
2
7
4
3
8
5
We will perform the following row operations on A:
Then A changes to U =
1
3
1
0
1
2
0
0
4
We may observe that U is an upper triangular matrix. Further, if we perform the 1st of the above row operation on I3, Then we get E1 =
1
0
0
-2
1
0
0
0
1
Now, if we perform the 2nd of the above row operation on I3, Then we get E2 =
1
0
0
0
1
0
-3
0
1
Also, if we if we perform the 3rd of the above row operation on I3, Then we get E3 =
1
0
0
0
1
0
0
1
1
Then, E3E2E1 A=U, where E3E2E1 and U are as above.
Further, if E3E2E1 A=U, then A = (E3E2E1)-1U = LU, where L = (E3E2E1)-1 =
1
0
0
2
1
0
3
-1
1
as E3E2E1 =
1
0
0
-2
1
0
-5
1
1
1
3
1
2
7
4
3
8
5
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