JEE Advanced · 2014 · 25 May 2014 · Paper 1 · Matrices

Let $M=\left(\begin{array}{cc}a& b\\ b& c\end{array}\right)$ be a symmetric matrix. For $M$ to be invertible, $\det (M)=ac-b^2\ne 0$. Option (A) and (B) are equivalent for a symmetric matrix and state that the first column equals the transpose of the second row, which is always true for symmetric matrices, so these don't guarantee invertibility. Option (C): if $M$ is diagonal with nonzero diagonal entries, then $b=0$ and $a,c\ne 0$, so $\det (M)=ac\ne 0$, making $M$ invertible. Option (D): if $ac$ is not a perfect square and $a,b,c$ are integers, then $ac-b^2\ne 0$ (since if $ac-b^2=0$, then $ac=b^2$ would be a perfect square), so $M$ is invertible. Thus (C) and (D) are correct.