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Manipal MET2015MathematicsDeterminants

Let D_k= | array ccc a & 2^k & 2¹⁶-1 b & 3 (4^k ) & 2 (4¹⁶-1 ) c & 7 (8^k ) & 4 (8¹⁶-1 ) array | , then the value of _ k=1 ¹⁶ D_k is

Options

  1. A0
  2. Ba+b+c
  3. Ca b+b c+c a
  4. DNone of these

Correct answer

A. 0

Step-by-step solution

We have, _ k=1 ¹⁶ D_k= | array ccc a & ( _ k=1 ¹⁶ 2^k ) & 2¹⁶-1 b & 3 ( _ k=1 ¹⁶ 4^4 ) & 2 (4¹⁶-1 ) C & 7 ( _ k=1 ¹⁶ 8^k ) & 4 (8¹⁶-1 ) array | _ k=1 ¹⁶ D_k= | array ccc a & 2 ( 2¹⁶-1 2-1 ) & 2¹⁶-1 b & 3 4 ( 4¹⁶-1 4-1 ) & 2 (4¹⁶-1 ) c & 7 8 ( 8¹⁶-1 8-1 ) & 4 (8¹⁶-1 ) array | _ k=1 ¹⁶ D_k= | array lll a & 2 (2¹⁶-1 ) & (2¹⁶-1 ) b & 4 (4¹⁶-1 ) & 2 (4¹⁶-1 ) c & 8 (8¹⁶-1 ) & 4 (8¹⁶-1 ) array | _ k=1 ¹⁶ D_k=2 | array ccc a & (2¹⁶-1 ) & (2¹⁶-1 ) b & 2 (4¹⁶-1 ) & 2 (4¹⁶-1 ) c & 4 (8¹⁶-1 ) & 4 (8¹⁶-1 ) array | [taking 2 com

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