JEE Advanced 2014 Mathematics 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 nonzero entries in the main diagonalCorrect
(D)the product of entries in the main diagonal of M is not the square of an integerCorrect

Answer: (C),(D)

Solution

Write with integers a, b, d; . (A) First column (a, b)ᵀ equals the transpose of the second row (b, d): a = b = d, so det = 0: not invertible. (B) Second row (b, d)

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Question ID VK-026293