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# Write Minors and Cofactors of the elements of following determinants: $(ii) \quad \begin{vmatrix} 1&0&4 \\ 3&5&-1 \\ 0&1&2 \end{vmatrix}$

Note: This is part 2 of a 2 part question, split as 2 separate questions here.

Toolbox:
• Minor of an element $a_{ij}$ of a determinant is the determinant obtained by deleting it $i^{th}$ row and $j^{th}$ column in which $a_{ij}$ lies.Minor of an element $a_{ij}$ is denoted by $M_{ij}$
• Cofactor of an element $a_{ij}$ denoted by $A_{ij}$ is defined by $A_{ij}=(-1)^{i+j}M_{ij}$,where $M_{ij}$ is minor of $a_{ij}$
We know that minor of the element $a_{ij}$ is $M_{ij}$,Where i is the row and j is the column.

We have $M_{11}=\begin{vmatrix}5 & -1\\1 & 2\end{vmatrix}=10-(-1)=11$

$M_{12}=\begin{vmatrix}3 & -1\\0 & 2\end{vmatrix}=6-0=6$

$M_{13}=\begin{vmatrix}3 & 5\\0 & 1\end{vmatrix}=3-0=3$

$M_{21}=\begin{vmatrix}0 & 4\\1 & 2\end{vmatrix}=-4$

$M_{22}=\begin{vmatrix}1 & 4\\0 & 2\end{vmatrix}=2-0=2$

$M_{23}=\begin{vmatrix}1 & 0\\0 & 1\end{vmatrix}=1-0=1$

$M_{31}=\begin{vmatrix}0 & 4\\5 & -1\end{vmatrix}=0-20=-20$

$M_{32}=\begin{vmatrix}1 & 4\\3 & -1\end{vmatrix}=-1-12=-13$

$M_{33}=\begin{vmatrix}1 & 0\\3& 5\end{vmatrix}=5-0=5$

$A_{11}=(-1)^{1+1}(11)=11.$

$A_{12}=(-1)^{1+2}(6)=-6.$

$A_{13}=(-1)^{1+3}(3)=3.$

$A_{21}=(-1)^{2+1}(-20)=-(-4)=4.$

$A_{22}=(-1)^{2+2}(2)=2.$

$A_{23}=(-1)^{2+3}(1)=-1.$

$A_{31}=(-1)^{3+1}(-20)=-20.$

$A_{32}=(-1)^{3+2}(-13)=-(-13)=13.$

$A_{33}=(-1)^{3+3}(5)=5.$