The basic idea here is to reduce a matrix into its components. These components are called minors, and the goal of this is to reduce the minor to a 2x2 form so that you can take the determinant with ab-cd where:

|a b|

|c d|

The Laplace expansion can occur along any column/row and equates the determinant of the entire matrix to the sum of the determinants of the minors. Each minor is multiplied by (-1) to the power of its indices. For example row one column one is (-1)^2 and so is positive, but row two column one is (-1)^3 which is negative

For example:

|a b c|

|d e f|

|g h i|

a

|e f|

|h i|

+

-b

|d f|

|g i|

+

c

|d e|

|g h|

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