$A = [a_{ij}] = \begin{bmatrix} 2 & -1 & 5 \\ 1 & 3 & 2 \\ 5 & 0 & 4 \end{bmatrix}$ and $C_{ij}$ is the cofactor of element $a_{ij}$, then the value of $a_{21} \cdot C_{11} + a_{22} \cdot C_{12} + a_{23} \cdot C_{13}$ is

$0$

The correct answer is Option (2) → 0 ##

Given, $A = [a_{ij}] = \begin{bmatrix} 2 & -1 & 5 \\ 1 & 3 & 2 \\ 5 & 0 & 4 \end{bmatrix}$

$a_{21} = 1, a_{22} = 3, a_{23} = 2$

$C_{11} = (-1)^{1+1} \begin{vmatrix} 3 & 2 \\ 0 & 4 \end{vmatrix} = (1)(12 - 0) = 12$

$C_{12} = (-1)^{1+2} \begin{vmatrix} 1 & 2 \\ 5 & 4 \end{vmatrix} = (-1)(4 - 10) = 6$

$C_{13} = (-1)^{1+3} \begin{vmatrix} 1 & 3 \\ 5 & 0 \end{vmatrix} = (1)(0 - 15) = -15$

Now, $a_{21}C_{11} + a_{22}C_{12} + a_{23}C_{13} = (1)(12) + (3)(6) + (2)(-15)$

$= 12 + 18 - 30$

$= 30 - 30 = 0$