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 nonzero 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)

the product of entries in the main diagonal of $M$ is not the square of an integer

Since, $M$ be a symmetric matrix.
$\therefore M=M^T$
Let $M=\left[\begin{array}{cc}{a}_1& a_2\\ a_2& a_3\end{array}\right],\text{where}{a}_1,a_2,a_3\in I$

Now, When the first column of $M$ is the transpose of the second row of $M$,
$\left[\begin{array}{c}{a}_1\\ a_2\end{array}\right]={\left[\begin{array}{cc}{a}_2& a_3\end{array}\right]}^T\Rightarrow a_1=a_2=a_3$
So, the matrix will be
$M=\left[\begin{array}{cc}{a}_1& a_1\\ a_1& a_1\end{array}\right]\Rightarrow |M|=0$
Therefore, $M$ become non-invertible.

When the second row of $M$ is the transpose of the first column of $M$,
$\left[\begin{array}{cc}{a}_2& a_3\end{array}\right]={\left[\begin{array}{c}{a}_1\\ a_2\end{array}\right]}^T\Rightarrow a_1=a_2=a_3$
So, the matrix will be
$M=\left[\begin{array}{cc}{a}_1& a_1\\ a_1& a_1\end{array}\right]\Rightarrow |M|=0$
Therefore, again $M$ become non-invertible.

When $M$ is a diagonal matrix with nonzero entries in the main diagonal,
$M=\left[\begin{array}{cc}{a}_1& 0\\ 0& a_3\end{array}\right]\Rightarrow |M|=a_1{a}_3\ne 0$
Therefore, in this case $M$ is invertible.

When the product of entries in the main diagonal of $M$ is not the square of an integer,
$M=\left[\begin{array}{cc}{a}_1& a_2\\ a_2& a_3\end{array}\right]\Rightarrow |M|=a_1{a}_3-(a_2{)}^2\ne 0$
Therefore, in this case $M$ is invertible.