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 $M=\begin{bmatrix}a & c \\ c & b\end{bmatrix}$ (where $\left.a, b, c \in I\right)$
then $\text{Det} M =a b - c ^{2}$
if $a = b = c , \text{Det}( M )=0$
if $c =0, a , b \neq 0, \text{Det}( M ) \neq 0$
if $a b \neq$ square of integer, $\text{Det}( M ) \neq 0$