Let $A = {\begin{pmatrix} a &b \\ c & d \end{pmatrix}} $
$A_{11} = d$,$A_{12} = c$,$A_{21} = b$,$A_{22} = a$
$\implies |A_{11}| = d$,$|A_{12}| = c$,$|A_{21}| = b$,$|A_{22}| = a$
As we can see $|A_{12}| = |A_{21}|$ if $c=b$
$\implies$ if matrix is ${\begin{pmatrix} a &c \\ c & d \end{pmatrix}} $
$\implies A$ should be a symmetric matrix.
$\therefore$ Option $C.$ is correct answer