Linear Algebra-Determinant Patterns

When solving for the determinant it is acceptable to use whatever pattern you want in order to find the solution. Normally when solving for the determinant we can use the first row in order to reduce the matrix into its 2x2 portions.

Example:
|1 2 3|
|4 5 6|
|7 8 9|
The normal formula for the determinant of a 3 by 3 matrix would be
det=1(5x9-6x8)-2(4x9-6x7)+3(4x8-5x7)
det=45-48-2(36-42)+3(32-35)
det=0

However we can solve this in a more arbitrary fashion utilizing a different row than the top row

det=-4(2x9-3x8)+5(1x9-3x7)-6(1x8-2x7)

det=-4(18-24)+5(9-21)-6(8-14)

det=0

Or by the last row

det=7(2x6-3x5)-8(1x6-3x4)+9(1x5-2x4)

det=7(12-15)-8(6-12)+9(5-8)

det=0

This can be used to make it easier to solve determinants by finding the simplest pattern to solve for the determinant.

|0 0 2 3 1|

|0 0 0 2 2|

|0 9 7 9 3|

|0 0 0 0 5|

|3 4 5 8 5|

-5

|0 0 2 3|

|0 0 0 2|

|0 9 7 9|

|3 4 5 8|

-5(2)

|0 0 2|

|0 9 7|

|3 4 5|

-5(2)(2)

|0 9|

|3 4|

-5(2)(2)(0x4-3x9)

det=540