Question

lf $A=\left\{\begin{array}{ccc}a& c& b\\ b& a& c\\ c& b& a\end{array}\right\}$ then the cofactor of $a_{32}$ in $A+A^T$ is

• A. $-\left(2a\right(b+c)-(b+c{)}^2)$
• B. $ac-b^2$
• C. $a^2-bc$
• D. $2a(a+c)-(a+c{)}^2$

Answer 1

Answer: (A)

$-\left(2a\right(b+c)-(b+c{)}^2)$

By property of transpose, we get
$A^T=\left[\begin{array}{ccc}a& b& c\\ c& a& b\\ b& c& a\end{array}\right]$
So, $A+A^T=\left[\begin{array}{ccc}2a& b+c& b+c\\ b+c& 2a& b+c\\ c+b& b+c& 2a\end{array}\right]$
So, $M_{32}=$ minor of $a_{32}=\left|\begin{array}{cc}2a& b+c\\ b+c& c+c\end{array}\right|=2a(b+c)-(b+c{)}^2$
$=2ab+2ac-b^2-c^2-2bc$
So, cofactor of $a_{32}=M_{32}(-1{)}^{3+2}$
$=-M_{32}$
$=-\left[2a\right(b+c)-(b+c{)}^2]$