Question

If $A=\left[a_{ij}\right]$ is a $2\times 2$ matrix, such that sum of co-factors of its elements is equal to the sum of elements of $A.$ Then which of the following can be matrix $A$ ?

• A. $\left[\begin{array}{cc}-1& 3\\ 0& 1\end{array}\right]$
• B. $\left[\begin{array}{cc}1& -4\\ 4& 2\end{array}\right]$
• C. $\left[\begin{array}{cc}3& 7\\ -7& -7\end{array}\right]$
• D. $\left[\begin{array}{cc}-7& -4\\ 5& 7\end{array}\right]$

Answer 1

Answer: (B), (C)

$\left[\begin{array}{cc}3& 7\\ -7& -7\end{array}\right]$

Let $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$
We know, $C_{ij}=(-1{)}^{i+j}{M}_{ij}$
$\Rightarrow C_{11}=d,C_{12}=-c,C_{21}=-b,C_{22}=a$
So, sum of co-factors is $a+d-(b+c)=a+b+c+d$ (given)
$\Rightarrow b+c=0$ (always)