Question

Let $M$ be a $2\times 2$ symmetric matrix with integer entries. Then, $M$ is invertible, if

• A. the first column of $M$ is the transpose of the second row of $M$
• B. the second row of $M$ is the transpose of the first column of $M$
• C. $M$ is a diagonal matrix with non-zero entries in the main diagonal
• D. the product of entries in the main diagonal of $M$ is not the square of an integer

Answer 1

Answer: (C), (D)

Let $M=\left[\begin{array}{cc}a& c\\ c& b\end{array}\right]$ (where $a,b,c\in I)$
then $\text{Det}M=ab-c^2$
if $a=b=c,\text{Det}(M)=0$
if $c=0,a,b\ne 0,\text{Det}(M)\ne 0$
if $ab\ne$ square of integer, $\text{Det}(M)\ne 0$