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)

The correct options are
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

For Matrix to be Invertible, determinant must not be equal to zero.

That is matrix should be non-singular

Let Matrix M$=\left[\begin{array}{cc}a& h\\ h& b\end{array}\right]$

Then determinant $=$ $ab-h^2$ which must not be equal to zero.

therefore ab not equal to $h^2$.

Therefore M is a diagonal Matrix with non-zero entries in the main diagonal.

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