Consider the matrix M = bmatrix 2 & -1 \ 1 & 0 bmatrix . Let p, q, r, s, a, b, c and d be integers such that M26 = bmatrix p & q \ r & s bmatrix and k=126 Mk = bmatrix a & b \ c & d bmatrix . Then which of the following statements is (are) TRUE ?
- There exists a 2 2 invertible matrix N with real entries such that MN = N bmatrix 1 & 1 \ 0 & 1 bmatrix
- The value of a is 378
- For any two given integers m and n, there exist unique integers x and y such that px + qy = m and rx + sy = n
- For each positive real number t, the system of linear equations (a + t)x + by = 1 cx + (d + t)y = -1 has a unique solution
Answer
(1) There exists a 2 2 invertible matrix N with real entries such that MN = N bmatrix 1 & 1 \ 0 & 1 bmatrix, (3) For any two given integers m and n, there exist unique integers x and y such that px + qy = m and rx + sy = n, (4) For each positive real number t, the system of linear equations (a + t)x + by = 1 cx + (d + t)y = -1 has a unique solution
Solution
Write M = I + K with K = bmatrix 1 & -1 \ 1 & -1 bmatrix, K2 = O, so Mk = I + kK.
(A) true: characteristic polynomial ( - 1)2 and M I, so M is similar to the Jordan block bmatrix 1 & 1 \ 0 & 1 bmatrix, i.e. N-1MN = J for some invertible N.
k=126 Mk = 26I + 351K = bmatrix 377 & -351 \ 351 & -325 bmatrix, so a = 377: (B) false.
(C) true: M26 = bmatrix 27 & -26 \ 26 & -25 bmatrix, = 1, so its inverse has integer entries and (x, y) is unique and integral.
(D) true: = (377 + t)(t - 325) + 3512 = t2 + 52t + 676 = (t + 26)2 > 0 for t > 0.
Answer: (A), (C), (D)