# Write Minors and Cofactors of the elements of following determinants: $\quad \begin{vmatrix} a&c \\ b&d \end{vmatrix}$

$\begin{array}{1 1} M_{12} = b, M_{21} = c, M_{22} = a \quad A_{11} = d, A_{12} = -b, A_{21} = -c, A_{22} = a \\ M_{12} = b, M_{21} = c, M_{22} = a \quad A_{11} = c, A_{12} = -c, A_{21} = -c, A_{22} = a \\ M_{12} = b, M_{21} = c, M_{22} =a \quad A_{11} = d, A_{12} = -c, A_{21} = -b, A_{22} = a \\ M_{12} = a, M_{21} = c, M_{22} = b \quad A_{11} = d, A_{12} = -b, A_{21} = -c, A_{22} = a \end{array}$

Toolbox:
• (i)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 $M_{ij}$
• (ii)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 $a_{ij}$
(ii)Given $\begin{vmatrix} a & c\\b & d\end{vmatrix}$

We know that minor of the element $a_{ij}$ is $M_{ij}$

Where i is the row and j is the column.

Here $a_{11}=a,$so $M_{11}$=Minor of $a_{11}$ =d

$M_{12}$=Minor of the element $a_{12}=b.$

$M_{21}$=Minor of the element $a_{21}=c.$

$M_{22}$=Minor of the element $a_{22}=a.$

Now the cofactor of $a_{ij}$ is $A_{ij}$,so

$A_{11}=(-1)^{1+1}\times M_{11}=(-1)^23\times d=d.$

$A_{12}=(-1)^{1+2}\times M_{12}=(-1)^3\times b=-b.$

$A_{21}=(-1)^{2+1}\times M_{21}=(-1)^3\times c=-c.$

$A_{22}=(-1)^{2+2}\times M_{22}=(-1)^4\times a=a.$