matrix 1:
      1     0     0
      0     1     0
      0     0     1
  determinant = 1, invertible
matrix 2:
      2    -3     1
      2     0    -1
      1     4     5
  determinant = 49, invertible
matrix 3:
      1     2     3
      4     5     6
      1     2     3
  determinant = 0, not invertible
matrix 4:
      2     1     1
      1     3     2
      1     0     0
  determinant = -1, invertible
