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# Write Minors and Cofactors of the elements of following determinants: $(i) \quad \begin{vmatrix} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{vmatrix}$

Note: This is part 1 of a 2 part question, split as 2 separate questions here.
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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 $M_{11}=\begin{vmatrix}1 & 0\\0 & 1\end{vmatrix}=1-0=1$

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

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

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

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

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

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

$M_{32}=\begin{vmatrix}1 & 0\\0 & 0\end{vmatrix}=0-0=0$

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

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

$A_{12}=(-1)^{1+2}(0)=0.$

$A_{13}=(-1)^{1+1}(0)=0.$

$A_{21}=(-1)^{2+1}(0)=0.$

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

$A_{23}=(-1)^{2+3}(0)=0.$

$A_{31}=(-1)^{3+1}(0)=0.$

$A_{32}=(-1)^{3+2}(0)=0.$

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

answered Feb 24, 2013
edited Feb 28, 2013