Use a determinant to determine whether the points (9, −7), (7, −9) and (12, −5) are collinear.

Three points are collinear if and only if the determinant of the matrix found by placing the x-coordinates in the first column, the y-coordinates in the second column, and one's in the third column is equal to zero.

So we have 

[  9     -7    1

   7     -9    1

  12    -5    1 ]

Re-write the first two rows

   9     -7    1    9    -7

   7     -9    1    7    -9

  12    -5    1   12   -5 

Now, take the determinant

[9(-9)(1) + (-7)(1)(12) + (1)(7)(-5) ] - [ (12)(-9)(1) + (-5)(1)(9) + (1)(7)(-7)] = 

[ - 81 -84  - 35] - [-108 - 45 - 49]  =

-200 - (-202)  =   

2

So......these points are not collinear..........