Question

$\huge \underline{\bf{Question :- }}$

Show that in the determinant of order 3 × 3

i) the sum of the product of elements of any row ( or column ) with their corresponding cofactors is the value of the determinant

ii) the sum of the product of elements of any row ( or column ) with the cofactor of the corresponding elements of any other row ( or column ) is zero​

Answer 1

$\huge{\blue{\boxed{\boxed{\boxed{\boxed{\bold{\red{Answer:}}}}}}}}$

i.) In the above attachment..

ii.) it will be zero

multiplying a row by the cofactors of any other row will mean that the row itself is duplicated on the determinant being evaluated. It is like calculating a determinant with two equal rows. And we know that a determinant with elementry row operations is the same determinant. So a determinant with two identical rows will be a determinant with a row replaced by different of those rows. Thus it will be zero.

Answer 2

${\huge\fbox\pink{A}\fbox\blue{N}\fbox\purple{S}\fbox\green{W}\fbox\red{E}\fbox\orange{R}\fbox{}\fbox\gray{}}$

i.) In the above attachment..

ii.) it will be zero

multiplying a row by the cofactors of any other row will mean that the row itself is duplicated on the determinant being evaluated. It is like calculating a determinant with two equal rows. And we know that a determinant with elementry row operations is the same determinant. So a determinant with two identical rows will be a determinant with a row replaced by different of those rows. Thus it will be zero.