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AP EAMCET202124 Aug 2021Morning ShiftMathematicsMatricesActual

Which of the following matrix has rank 3?

Options

  1. A[ array ccc 10 & 11 & 12 11 & 12 & 13 12 & 13 & 14 array ]
  2. B[ array ccc 0 & -51 & 101 51 & 0 & -581 -101 & 581 & 0 array ]
  3. C[ array ccc 0 & 1 & 2 -1 & 0 & 5 -2 & 7 & 0 array ]
  4. D[ array lll 1 & 2 & 3 2 & 4 & 6 3 & 6 & 9 array ]

Correct answer

C. [ array ccc 0 & 1 & 2 -1 & 0 & 5 -2 & 7 & 0 array ]

Step-by-step solution

Option (a) Let A= [ array lll 10 & 11 & 12 11 & 12 & 13 12 & 13 & 14 array ] Applying R₁ R₁-R₂, R₂ R₂-R₃ [ array ccc -1 & -1 & -1 -1 & -1 & -1 12 & 13 & 14 array ] Applying R₂ R₂-R₁, R₃ R₃+12 R₁ , [ array ccc -1 & -1 & -1 0 & 0 & 0 0 & 1 & 2 array ] Applying R₂ R₃ [ array ccc -1 & -1 & -1 0 & 1 & 2 0 & 0 & 0 array ] this is echelon form Rank of matrix A=2 Option (b) B= [ array ccc 0 & -51 & 101 51 & 0 & -581 -101 & 581 & 0 array ] B^T=-B, B is skew-symmetric matrix. Rank of skew-symmetric matrix is always an even n

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