### Row Echelon Form

Exercise Explain why the following are not row echelon forms by showing how the shape can be improved.

 [ 0 * # # # ] 0 0 * # # * # # # #
 [ 0 * # # # ] 0 0 * # # 0 * # # #
 [ 0 0 * # # ] 0 * # 0 0 * 0 0 0 0
 [ # # # # * ] # # # * 0 # # * 0 0

Exercise Compute the (reduced) row echelon form and list the pivot columns.

 [ 0 ]
 [ 0 1 ]
 [ 1 2 ]
 [ 2 1 ]
 [ 0 ] 1
 [ 1 ] 2
 [ 2 ] 1
 [ 0 -1 ] 0 0 0 0
 [ 0 0 0 1 ] 0 0 1 0 0 1 0 0 1 0 0 0
 [ 1 2 ] 0 0 0 0 0 0
 [ 0 1 0 2 0 ] 0 0 1 0 2 0 0 0 1 0 0 0 0 0 0
 [ 1 0 0 0 ] -1 1 0 0 0 -1 1 0 0 0 -1 1
 [ 1 2 1 1 ] 2 1 1 0 0 3 1 0 3 0 1 0
 [ 1 0 1 2 ] 1 4 -3 -2 1 -1 2 3
 [ 2 -3 6 2 5 ] -2 3 -3 -3 -4 4 -6 9 5 9 -2 3 3 -4 1
 [ 1 -2 1 1 2 ] -1 3 0 2 -2 0 1 1 3 4 1 2 5 13 5
 [ 1 -1 0 1 ] -2 3 1 2 1 0 1 5 1 2 3 13 2 -2 4 5
 [ 1 2 3 0 1 1 ] 0 3 2 1 -1 0 -1 1 2 -2 1 0 1 -2 -2 1 0 0 4 1 3 2 2 1

Exercise Use different row operations to find a (reduced) row echelon forms for the matrices in this example and this example. Compare your result with the ones obtained in the examples.

Exercise List all the shapes of the row echelon forms of 2 by 2 matrices? What about the shapes of reduced row echelon forms? What about the other sizes (2 by 3, 3 by 2, etc.)?

Exercise Check out the examples of row echelon forms to convince yourself that

number of pivots = number of pivot rows = number of pivot columns.

Can you provide the reason in general?

Exercise Show that for a 3 by 3 matrix, all rows are pivot ⇔ all columns are pivot? What about general n by n matrices?