Question

Let A be the $2\times 2$ matrices given by $A=\left[a_{jj}\right]where{a}_{jj}ϵ\{0,1,2,3,4\}$ such that $a_{11}+a_{12}+a_{21}+a_{22}=4$, then the number of matrices A such that A is either symmetric or skew-symmetric or both and det(A) is divisible by 2 are

• A. 5
• B. 3
• C. 7
• D. 8

Answer 1

The correct option is A 5

There will not be any skew-symmetric matrix because no element is negative and sum of elements is 4. For symmetric matrix, pair of conjugate elements must be same.
$Type-I\left[\begin{array}{cc}4& 0\\ 0& 0\end{array}\right]\left[\begin{array}{cc}0& 0\\ 0& 4\end{array}\right]Type-II\left[\begin{array}{cc}3& 0\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& 3\end{array}\right]Type-III\left[\begin{array}{cc}2& 1\\ 1& 0\end{array}\right]\left[\begin{array}{cc}0& 1\\ 1& 2\end{array}\right]Type-IV\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right]\left[\begin{array}{cc}0& 2\\ 2& 0\end{array}\right]Type-V\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$
There are 5 symmetric matrices A such that det(A) is divisible by 2.