Question

If $A=\left\{\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right\}$ then cofactor of $a_{21}$ is:

• A. $b^2-$ ac
• B. $ac-b^2$
• C. $a^2-bc$
• D. $bc-a^2$

Answer 1

The correct option is B $ac-b^2$

By property of co-factor of matrix(2),
cofactor of $a_{ij}=M_{ij}\times (-1{)}^{i+j}$
$\therefore$ minor of $a_{21}=\left|\begin{array}{cc}{a}_{12}& a_{13}\\ a_{32}& a_{33}\end{array}\right|=\left|\begin{array}{cc}b& c\\ a& b\end{array}\right|$
$M_{21}=b^2-ac$
So, co factor of $a_{21}=M_{21}\times (-1{)}^{1+2}=-M_{21}$
$=ac-b^2$