Match the following elements of $\begin{bmatrix}1 & -1 & 0 \\0 & 4 & 2\\3 & -4 & 6\end{bmatrix}$ with their co-factors.

Choose the correct answer from the options given below :

A-III,B-IV,C-II,D-I

The correct answer is option (1) → A-III,B-IV,C-II,D-I

$C_{11}=(-1)^2\begin{vmatrix}4&2\\-4&6\end{vmatrix}=32$, $C_{12}=(-1)^3\begin{vmatrix}0&2\\3&6\end{vmatrix}=6$

$C_{13}=(-1)^{1+3}\begin{vmatrix}0&4\\3&-4\end{vmatrix}=-122$, $C_{21}=(-1)^3\begin{vmatrix}-1&0\\-4&6\end{vmatrix}=6$

$C_{22}=(-1)^4\begin{vmatrix}1&0\\3&6\end{vmatrix}=6$, $C_{23}=(-1)^5\begin{vmatrix}1&-1\\3&-4\end{vmatrix}=1$

$C_{31}=(-1)^4\begin{vmatrix}-1&0\\4&2\end{vmatrix}=-2$, $C_{32}=(-1)^3\begin{vmatrix}1&0\\0&2\end{vmatrix}=-2$

$C_{33}=(-1)^6\begin{vmatrix}1&-1\\0&4\end{vmatrix}=4$

so (A) 6 → (III) 4

(B) 3 → (IV) -2

(C) 2 → (II) 1

(D) -1 → (I) 6