Let A be a 2 × 2 matrix of the form A = a b 1 1 , where a , b are integers and - 50 ≤ b ≤ 50 . The number of such matrices A such that A - 1 , the inverse of A , exists and A - 1 contains only integer entries is

Question

Let A be a 2 × 2 matrix of the form A = a b 1 1 , where a , b are integers and - 50 ≤ b ≤ 50 . The number of such matrices A such that A - 1 , the inverse of A , exists and A - 1 contains only integer entries is, 101, 200, 202, 101 2

MARKS App · Accepted Answer

≠ 0 ⇒ a - b ≠ 0 ⇒ a ≠ b . . . ( i ) Also, A − 1 = 1 a − b 1 − 1 − b a T = 1 a − b 1 − b − 1 a Thus, a - b = 1 or - 1 . . . ii So, required number of pairs ( a , b ) is 101 × 2 = 202

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