SOLUTION: What is the rank of the following matrix? |18 -27 -3| |-2 3 -5| |2 -3 8|

Its rank is the number of rows that do not
contain all 0's in the row-reduced-echelon form,
so we get it in its row-reduced-echelon form


[[18 -27 -3]
 [-2   3 -5]
 [ 2  -3  8]]

Add the second row to the third row:

[[18 -27 -3]
 [-2   3 -5]
 [ 0   0  3]]

Multiply the 2nd row by 9 

[[ 18 -27  -3]
 [-18  27 -45]
 [  0   0   3]]


Add the 1st row to the 2nd row:

 [[18 -27  -3]
  [ 0   0 -48]
  [ 0   0   3]]

Multiply the bottom row by 16

 [[18 -27  -3]
  [ 0   0 -48]
  [ 0   0  48]]

Add the 2nd row to the 3rd row

 [[18 -27  -3]
  [ 0   0 -48]
  [ 0   0   0]]

Divide the 2nd row through by -48

 [[18 -27  -3]
  [ 0   0   1]
  [ 0   0   0]]

Multiply the 2nd row by 3

 [[18 -27  -3]
  [ 0   0   3]
  [ 0   0   0]]

Add the 2nd row to the 1st row:

 [[18 -27  0]
  [ 0   0  3]
  [ 0   0  0]]

Divide the top row through by 18
Divide the 2nd row by 3

 [[1 -1.5  0]
  [0    0  1]
  [0    0  0]]

That's the row-reduced echelon form.  It has 2 rows which
aren't all 0's, so its rank is 2 and it has 1 row that contains
only 0's, so its nullity is 1.

Edwin