Find minors and cofactors of the elements of the determinant`|2-3 5 6 0 4 1 5-7|`and verify that `a_(11)A_(31)+a_(12)A_(32)+a_(13)A_(33)=0`

Explanation

`Delta=|[2,-3, 5],[ 6, 0, 4],[ 1, 5,-7]|`

`Delta=2|[0,4],[5,-7]|-(-3)|[6,4],[1,-7]|+5|[6,0],[1,5]|`

`Delta=2[0(-7)-5(4)]+3[6(-7)-4(1)]+5[6(5)-0(1)]`

`Delta=2(0-20)+3(-42-4)+5(30-0)`

`Delta=2(-20)+3(-46)+5(30)=-40-138+150`

`Delta=-28`

Interchanging `R_2` and `R_3`
i.e. `R_2 leftrightarrow R_3`,
We get

`Delta_1=|[2,-3,5],[1,5,-7],[6,0,4]|`

`Delta_1=2|[5,-7],[0,4]|-(-3)|[1,-7],[6,4]|+5|[1,5],[6,0]|`

`Delta_1=2[5(4)-0(-7)]+3[1(4)-6(-7)]+5[1(0)-6(5)]`

`Delta_1=2[20-0]+3[4+42]+5[0-30]`

`Delta_1=2[20]+3[46]+5[-30]=40+138-150`

`Delta_1=28`

`Delta_1=-(-28)=-Delta`

`therefore` `Delta_1=-Delta`

Hence property 2 is verified
i.e. if any two rows (or columns)of a determinant are interchanged, then sign of determinant changes.