If $a_{ij}$ represents the elements of $i^{th}$ row and $j^{th}$ column and $A_{ij}$ is the corresponding cofactor, then for the matrix $A=\begin{bmatrix} 1 & -1 & 0\\2 & 0 & 1\\3 & 4 & 2\end{bmatrix}$ the value of $a_{11}A_{21}+a_{12}A_{22}+a_{12}A_{22}+a_{13}A_{23}$ is :

-5

The correct answer is Option (2) → -5

$A=\begin{bmatrix} 1 & -1 & 0\\2 & 0 & 1\\3 & 4 & 2\end{bmatrix}$

$C_{11}=-4,C_{12}=-1,C_{13}=8$

$C_{21}=2,C_{22}=2,C_{23}=-7$

$C_{31}=-1,C_{32}=-1,C_{33}=2$

so $Adj\,A=\begin{bmatrix}-4&-1&8\\2&2&-7\\-1&-1&2\end{bmatrix}^T=\begin{bmatrix}-4&2&-1\\-1&2&-1\\8&-7&2\end{bmatrix}$

$a_{11}A_{21}+a_{12}A_{22}+a_{12}A_{22}+a_{13}A_{23}=1×(-1)+-1×2+-1×2+0×-1$

$=-1-2-2=-5$