Question:

Let M be a 2x2 symmetric matrix with integer entries. Then M is invertible if:

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

The second row of M is the transpose of the first column of M

The first column of M is the transpose of the second row of M

M is a diagonal matrix with non-zero entries in the main diagonal

Solution:

For Matrix to be Invertible, determinant must not be equal to zero. That is matrix should be non-singular. Let Matrix M = [[a, h], [h, b]]. 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.