For $A=\left[\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right]$ if $M_{i j}$ and $A_{i j}$ are the minor and the cofactor of $a_{i j}$ respectively, then |A| equals to :

(A) $a_{11} A_{11}+a_{12} A_{12}+a_{13} A_{13}$

(B) $a_{21} A_{12}+a_{22} A_{22}+a_{23} A_{32}$

(C) $a_{11} M_{11}-a_{12} M_{12}+a_{13} M_{13}$

(D) $a_{11} M_{11}+a_{12} M_{12}+a_{13} M_{13}$

Choose the correct answer from the options given below :

(A), (C) Only

$A=\left[\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right]$

$M_{ij}$ = determinant of matrix formed by eliminating i^th row and j^th column

$A_{ij} = (-1)^{i+j} M_{ij}$   →  Relation between minor and cofactor

sum of products of corresponding 

a_ij and A_ij = |A|

⇒  |A| = a₁₁ A₁₁ + a₁₂ A₁₂ + a₁₃ A₁₃

=  a₁₁(-1)¹⁺¹ M₁₁ + a₁₂(-1)¹⁺² M₁₂ + a₁₃(-1)¹⁺³ M₁₃

|A| = a₁₁ M₁₁ - a₁₂ M₁₂ + a₁₃ M₁₃

Option → (2)   (A), (C)