Question

If $A=\left[a_{ij}\right]$ is a $2\times 2$ matrix. Then which of the following statement is always true :

• A. Sum of co-factors of elements of $A$ is equal to sum of its elements
• B. Sum of co-factors of elements of $A$ is equal to the difference in sum of its row elements
• C. Product of co-factors of elements of $A$ is equal to the product of its elements
• D. Product of co-factors of elements of $A$ is equal to the product of square of its elements

Answer 1

The correct option is C Product of co-factors of elements of $A$ is equal to the product of its elements

Let $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$
Then,
$M_{11}=d,M_{12}=c,M_{21}=b,M_{22}=a$
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 $=a+d-(b+c)\ne a+b+c+d$
Product of co-factors $=abcd$