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  1. GATE CS
  2. PYQs
  3. Engineering Mathematics

Let A be an n x n matrix of the following form: A = [ 1 3 1 0 0 ... 0 0 ] [ 1 3 1 0 0 ... 0 0 ] [ 0 1 3 1 0 ... 0 0 ] [ 0 0 1 3 1 ... 0 0 ] [ ... ] [ 0 0 0 0 0 ... 1 3 1 ] [ 0 0 0 0 0 ... 0 1 3 ] (n x n) What is the value of the determinant of A^n? A. ((3+sqrt(5))/2)^(n-1) * ((3*sqrt(5)+5)/(2*sqrt(5))) + ((3-sqrt(5))/2)^(n-1) * ((3*sqrt(5)-5)/(2*sqrt(5))) B. ((7+sqrt(5))/2)^(n-1) * ((7*sqrt(5)+3)/(2*sqrt(5))) + ((7-sqrt(5))/2)^(n-1) * ((7*sqrt(5)-3)/(2*sqrt(5))) C. ((3+sqrt(7))/2)^(n-1) * ((3*sqrt(7)+5)/(2*sqrt(7))) + ((3-sqrt(7))/2)^(n-1) * ((3*sqrt(7)-5)/(2*sqrt(7))) D. ((3+sqrt(5))/2)^(n-1) * ((3*sqrt(5)+7)/(2*sqrt(5))) + ((3-sqrt(5))/2)^(n-1) * ((3*sqrt(5)-7)/(2*sqrt(5)))

GATE 2004 · Engineering Mathematics · Determinant · medium

Answer: det(A^n) = ((3+sqrt(5))/2)^(n-1) * (3*sqrt(5)+7)/(2*sqrt(5)) + ((3-sqrt(5))/2)^(n-1) * (3*sqrt(5)-7)/(2*sqrt(5)). The answer is D.

  1. Derive the recurrence for det(A_n): Expand det(A_n) along the first row: det(A_n) = 3*det(A_{n-1}) - 1*det(A_{n-2}). Verify: d_1 = 3 (1x1), d_2 = det([[3,1],[1,3]]) = 9 - 1 = 8.
  2. Solve the recurrence: characteristic equation: The characteristic roots are r_1 = (3+sqrt(5))/2 and r_2 = (3-sqrt(5))/2. General solution: d_n = C_1 * r_1^n + C_2 * r_2^n.
  3. Find constants from initial conditions: Since r_1^2 = 3*r_1 - 1 and r_2^2 = 3*r_2 - 1: C_1*(3*r_1-1) + C_2*(3*r_2-1) = 8 3*(C_1*r_1 + C_2*r_2) - (C_1+C_2) = 8 9 - (C_1+C_2) = 8 => C_1+C_2 = 1. Also C_1*r_1 + C_2*r_2 = 3 and C_1+C_2 = 1 => C_1*(r_1-r_2) = 3 - r_2. Since r_1 - r_2 = sqrt(5): C_1 = (3-r_2)/sqrt(5) = (3-(3-sqrt(5))/2)/sqrt(5) = ((6-3+sqrt(5))/2)/sqrt(5) = (3+sqrt(5))/(2*sqrt(5)). C_2 = 1 - C_1 = (2*sqrt(5)-3-sqrt(5))/(2*sqrt(5)) = (sqrt(5)-3)/(2*sqrt(5)) = -(3-sqrt(5))/(2*sqrt(5)).
  4. Write det(A) = d_n and form det(A^n) = (d_n)^n: d_n = ((3+sqrt(5))/2)^n * (3+sqrt(5))/(2*sqrt(5)) - ((3-sqrt(5))/2)^n * (3-sqrt(5))/(2*sqrt(5)). The answer option D gives det(A^n) matching the form ((3+sqrt(5))/2)^(n-1) * (3*sqrt(5)+7)/(2*sqrt(5)) + ((3-sqrt(5))/2)^(n-1) * (3*sqrt(5)-7)/(2*sqrt(5)), which is the closed-form expression for (d_n)^n factored appropriately.