Find the determinant

Question

|2 0 0 0|

|-1 1 0 0|

|0 0 -4 0|

|1 -1 1 -1|

Answer 1

|2 0 0 0|
|-1 1 0 0|
|0 0 -4 0|
|1 -1 1 -1|

take the top row. Each column in top row is multiplied by the determinant of the 3x3 sub matrix which does not include the top row and this elements column. Every other term is alternately multiplied by plus or minus one. Starting with plus. 

=2•
| 1 0 0|
| 0 -4 0|
|-1 1 -1|

-0•

|-1 0 0|
|0 -4 0|
|1 1 -1|

+0•
|-1 1 0|
|0 0 0|
|1 -1 -1|

-0•
|-1 1 0|
|0 0 -4|
|1 -1 1|

 

=2•
| 1 0 0|
| 0 -4 0|
|-1 1 -1|

 

=2[(1)(-4)(-1)+0+0-[0+0+0]]

The 3x3 determinate is calculated multiplying the left tilting diagonal elements plus the diagonals to the right where the matrix wraps so each of the columns contribute the product of a three element  diagonal. Next  three right leaning diagonals are subtracted. Each of the six diagonals are the product of three elements. 

=8