If Delta = begin{vmatrix} a_{11} & a_{12} & a | vedclass

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(B) The value of a determinant $\Delta$ is equal to the sum of the products of elements of any row (or column) with their corresponding cofactors. For

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$a_{11} A_{11} + a_{21} A_{21} + a_{31} A_{31}$

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(B) The value of a determinant $\Delta$ is equal to the sum of the products of elements of any row (or column) with their corresponding cofactors. For any row $i$,$\Delta = \sum_{j=1}^{3} a_{ij} A_{ij} = a_{i1} A_{i1} + a_{i2} A_{i2} + a_{i3} A_{i3}$. For any column $j$,$\Delta = \sum_{i=1}^{3} a_{ij} A_{ij} = a_{1j} A_{1j} + a_{2j} A_{2j} + a_{3j} A_{3j}$. Looking at the options: Option $A$: $a_{11} A_{31} + a_{12} A_{32} + a_{13} A_{33}$ is the sum of products of elements of row $1$ with cofactors of row $3$,which equals $0$. Option $B$: $a_{11} A_{11} + a_{21} A_{21} + a_{31} A_{31}$ is the sum of products of elements of column $1$ with their corresponding cofactors,which is equal to $\Delta$. Option $C$: $a_{21} A_{11} + a_{22} A_{12} + a_{23} A_{13}$ is the sum of products of elements of row $2$ with cofactors of row $1$,which equals $0$. Option $D$: This is a mix of elements and cofactors that does not follow the expansion rule. Thus,the correct option is $B$.

Element	Co-factor
$A$. $-1$	$(1)$ $-2$
$B$. $1$	$(2)$ $32$
$C$. $3$	$(3)$ $4$
$D$. $6$	$(4)$ $6$
	$(5)$ $-6$

If $\Delta = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix}$ and $A_{ij}$ is the cofactor of $a_{ij}$,then the value of $\Delta$ is given by:

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