_{21}and E

_{32}and E

_{43}. What are those matrices?

This 4 by 4 matrix will need elimination matrices E_{21} and E_{32} and E_{43}. What are those matrices?

A=[2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2, -1, 0, 0, -1, 2].

The first row is 2, -1, 0, 0.

The second row is -1, 2, -1, 0.

The third row is 0, -1, 2, -1.

The fourth row is 0, 0, -1, 2.

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If you left-multiply

[ 2 -1 0 0]

[ -1 2 -1 0]

[ 0 -1 2 -1]

[ 0 0 -1 2]

by

[ 1 0 0 0]

[ 1/2 1 0 0]

[ 0 0 1 0]

[ 0 0 0 1]

, then you get

[ 2 -1 0 0]

[ 0 3/2 -1 0]

[ 0 -1 2 -1]

[ 0 0 -1 2]

, which would eliminate the first variable from all but the first equation of the system

AX = Y

, for whatever given column vector Y. So

[ 1 0 0 0]

[ 1/2 1 0 0]

[ 0 0 1 0]

[ 0 0 0 1]

[ 1/2 1 0 0]

[ 0 0 1 0]

[ 0 0 0 1]

must be E_{21}. Next, you would need to eliminate the second variable from all but the first two equations. Could you come up with the matrix E_{32} by which to left-multiply E_{21}A?

Finally, find E_{43} by which to left-multiply E_{32}E_{21}A to get upper-triangular matrix E_{43}E_{32}E_{21}A.

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## Comments

Ayou gave me is multiplied on the left by the elimination matrix I came up with. That adds 1/2 times the first row ofAto the second row ofA, getting rid of that first -1.