Question

Let M be a 2 × 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 is

(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

⇒ Not invertible. Therefore, (A) is false.

(B) [b, d] = [a, b] ⇒ a =b = d

⇒ |M| = 0 ⇒ Not invertible. Therefore, (B) is false.

(C) If M is a diagonal matrix, then M= [(a, 0), (0, d)] ⇒ |M| = ad ≠ 0

 ⇒ M invertible. Therefore, (C) is correct .

(D) Given ad ≠ b². Now |M| = ad − b² ≠ 0 for M to be invertible. Therefore, (D) is true.