REDUCING MATRICES

A matrix that reduces to this Can readily be interpreted

1. Place 1 in the upper left If you follow this order, you will not waste time or go around in circles. 2. Place 0’s in the column under the red 1 3. Place second 1 on the diagonal 4. Place 0’s in the column under the blue 1 5. Place third 1 on the diagonal 6. Place 0 in the column under the green 1 7. Place fourth 1 on the diagonal 8. A 0 in the upper right triangle can be placed at any point after the 1 and the lower 0’s have been placed in its column

Consider this system. First extract the coefficient matrix

+2 1 -1 0 -1 L1 becomes: First we want a 1 in this position. Replace L1 with L1 + 2 L2 1 -1 0 -1 L1 becomes:

+2 L2 becomes 0 -1 -1 -4 Second we want 0’s in these positions. Replace L2 with L2 + 2 L1 +2 L2 becomes 0 -1 -1 -4 Second we want 0’s in these positions.

+1 L2 becomes 0 -1 -1 -4 L3 becomes 0 0 1 1 Replace L3 with L3 + 1 L1 +1 L2 becomes 0 -1 -1 -4 Second we want 0’s in these positions. L3 becomes 0 0 1 1 Replace L2 with L2 + 2 L1

Replace L2 with -1L2 L2 becomes 0 1 1 4 Third we want a 1 in this position.

Fourth we have a 0 in this position. And a 1 in this position

-1 L2 becomes 0 1 0 3 Finally we need 0’s in these positions Replace L2 with L2 + -1 L3 -1

L1 becomes 1 0 0 2 Replace L1 with L1 + 1 L2 +1 L2 becomes 0 1 0 3