Question

Write the minors and cofactors of each element of the first column of the following matrices :
$A=\left[\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right]$

Answer 1

$\Rightarrow$ Let $M_{ij}$ and $C_{ij}$ represents the minor and co-factor of an element, where $i$ and $j$ represent the row an column.

$\Rightarrow$ The minor of matrix can be obtained for particular element by removing the row and column where the element is present. Then finding the absolute value of the matrix newly formed.

$\Rightarrow$ $M_{11}=\left|\begin{array}{cc}b& f\\ f& c\end{array}\right|=bc-f^2$

$\Rightarrow$ $M_{21}=\left|\begin{array}{cc}h& g\\ f& c\end{array}\right|=hc-fg$

$\Rightarrow$ $M_{31}=\left|\begin{array}{cc}h& g\\ b& f\end{array}\right|=hf-gb$

$C_{ij}=(-1{)}^{i+j}\times M_{ij}$

$\Rightarrow$ $C_{11}=(-1{)}^{1+1}\times (bc-f^2)=bc-f^2$

$\Rightarrow$ $C_{21}=(-1{)}^{2+1}\times (hc-fg)=-(hc-fg)=fg-hc$

$\Rightarrow$ $C_{31}=(-1{)}^{3+1}\times (hf-gb)=hf-gb$

$\Rightarrow$ $|A|=(a_{11}\times C_{11})+(a_{21}\times C_{21})+(a_{31}\times c_{31})$

$=a\times (bc-f^2)+h(fg-hc)+g(hf-gb)$

$=abc-a{f}^2+fgh-c{h}^2+fgh-b{g}^2$

$=abc+2fgh-a{f}^2-b{g}^2-c{h}^2$