# If and Aij is Cofactors of aij, then value of Δ is given by

Question:

If $\Delta=\left|\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{21} & a_{23} & a_{23}\end{array}\right|$ and $\mathrm{A}_{i j}$ is Cofactors of $a_{i j}$, then value of $\Delta$ is given by

(A) $\quad a_{11} \mathrm{~A}_{31}+a_{12} \mathrm{~A}_{32}+a_{13} \mathrm{~A}_{33}$

(B) $a_{11} \mathrm{~A}_{11}+a_{12} \mathrm{~A}_{21}+a_{13} \mathrm{~A}_{31}$

(C) $a_{21} \mathrm{~A}_{11}+a_{22} \mathrm{~A}_{12}+a_{23} \mathrm{~A}_{13}$

(D) $a_{11} \mathrm{~A}_{11}+a_{21} \mathrm{~A}_{21}+a_{31} \mathrm{~A}_{31}$

Solution:

We know that:

$\Delta=$ Sum of the product of the elements of a column (or a row) with their corresponding cofactors

$\therefore \Delta=a_{11} \mathrm{~A}_{11}+a_{21} \mathrm{~A}_{21}+a_{31} \mathrm{~A}_{31}$

Hence, the value of Δ is given by the expression given in alternative D.

The correct answer is D.