 Hello and welcome to the session. Let us understand the following problem today. Using cofactors of elements of third column evaluate, delta is equal to 1xyz1yz1zxy. Now let us write the solution. First let us find the cofactors of this row, this is a third row. So cofactor of A31 which is nothing but capital A31 which is equal to minus 1 to the power 3 plus 1 into determinant xyzyzx which is equal to zx squared minus zy squared. Now cofactor of A32 which is equal to capital A32 which is equal to minus 1 to the power 3 plus 2 into determinant 1yz1zx which is equal to minus into zx minus zy which is equal to zy minus zx. Now cofactor of A33 is equal to capital A33 which is equal to minus 1 to the power 3 plus 3 into determinant 1xy which is equal to y minus x. Now let us evaluate the determinant now which is equal to A31 A31 plus A32 A32 plus A33 A33 where small A31 A32 A33 are the elements of the determinants and capital A31 capital A32 capital A33 are the cofactors of the determinant. Now which is equal to 1 into zx squared minus zy squared plus z into zy minus zx plus xy into y minus x. Now taking z common we get x squared minus y squared plus again taking z common from here we get y minus x plus xy into y minus x. Now this x squared minus y squared can be written as z into x minus y into x plus y plus z squared into y minus x plus xy into y minus x. Now taking x minus y as common so we are left with z into x plus y minus z squared minus xy which is equal to x minus y into zx plus zy minus x squared minus xy which is equal to x minus y into z into x minus z plus y into z minus x. We have combined these two terms and then these two terms and then took the common so we got this expression. Now which is equal to x minus y into minus z into z minus x plus y into z minus x which is equal to x minus y y minus z into z minus x. Therefore our required answer is x minus y into y minus z into z minus x. I hope you understood the problem. Bye and have a nice day.