Q27 / 60 · JEE Advanced 2014 (Paper 1) · Mathematics

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
Correct answer
C
M is a diagonal matrix with non - zero entries in the main diagonal
Correct answer
D
The product of entries in the main diagonal of M is not the square of an integer

Solution

Let

M = [\begin{matrix} a & c \\ c & b \end{matrix}]

(where a, b, c

\in I

)
Then Det

M = a b - c^{2}

a = b = c, D e t (M) = 0

c = 0, a, b \neq 0, D e t (M) \neq 0

a b \neq

square of integer,

D e t (M) \neq 0

Hence, the correct answer is options: 'C', 'D'.