Write Minors and Cofactors of the elements of following determinants:
Question:

Write Minors and Cofactors of the elements of following determinants:

(i) $\left|\begin{array}{rr}2 & -4 \\ 0 & 3\end{array}\right|$

(ii) $\left|\begin{array}{ll}a & c \\ b & d\end{array}\right|$

Solution:

(i) The given determinant is $\left|\begin{array}{rr}2 & -4 \\ 0 & 3\end{array}\right|$.

Minor of element $a_{i j}$ is $M_{i j}$.

$\therefore \mathrm{M}_{11}=$ minor of element $a_{11}=3$

$M_{12}=$ minor of element $a_{12}=0$

$M_{21}=$ minor of element $a_{21}=-4$

$M_{22}=$ minor of element $a_{22}=2$

Cofactor of $a_{i j}$ is $A_{i j}=(-1)^{\gamma+j} M_{i j}$.

$\therefore A_{11}=(-1)^{1+1} M_{11}=(-1)^{2}(3)=3$

$A_{12}=(-1)^{1+2} M_{12}=(-1)^{3}(0)=0$

$A_{21}=(-1)^{2+1} M_{21}=(-1)^{3}(-4)=4$

$A_{22}=(-1)^{2+2} M_{22}=(-1)^{4}(2)=2$

(ii) The given determinant is $\left|\begin{array}{ll}a & c \\ b & d\end{array}\right|$.

Minor of element $a_{i j}$ is $M_{i j}$.

$\therefore \mathrm{M}_{11}=$ minor of element $a_{11}=d$

$M_{12}=$ minor of element $a_{12}=b$

$M_{21}=$ minor of element $a_{21}=c$

$\mathrm{M}_{22}=$ minor of element $\mathrm{a}_{22}=\mathrm{a}$

Cofactor of $a_{i j}$ is $A_{i j}=(-1)^{j+j} M_{i j}$.

$\therefore \mathrm{A}_{11}=(-1)^{1+1} \mathrm{M}_{11}=(-1)^{2}(d)=d$

$\mathrm{~A}_{12}=(-1)^{1+2} \mathrm{M}_{12}=(-1)^{3}(b)=-b$

$\mathrm{~A}_{21}=(-1)^{2+1} \mathrm{M}_{21}=(-1)^{3}(c)=-c$

$\mathrm{~A}_{22}=(-1)^{2+2} \mathrm{M}_{22}=(-1)^{4}(a)=a$