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BITSAT2024MathematicsMatricesActual

If A= [ array cc 1 & 0 0 & -1 array ], P= [ array ll 1 & 1 0 & 1 array ] and X=A P A^T , then A^T X⁵⁰ A=

Options

  1. A[ array ll 0 & 1 1 & 0 array ]
  2. B[ array cc 2 & 1 0 & -1 array ]
  3. C[ array cc 25 & 1 1 & -25 array ]
  4. D[ array cc 1 & 50 0 & 1 array ]

Correct answer

D. [ array cc 1 & 50 0 & 1 array ]

Step-by-step solution

Since A A^T=I therefore matrix A is orthogonal matrix. Now, A^T X⁵⁰ A=A^T X⁴⁹ (A P A^T ) A=A^T X⁴⁹ A P (A^T A )=A^T X⁴⁹ A P=A^T X⁴⁸ (A P A^T ) A P=A^T X⁴⁸ A P^2 . =A^T A P⁵⁰=I P⁵⁰=P⁵⁰ P= [ array ll 1 & 1 0 & 1 array ] P^2= [ array ll 1 & 2 0 & 1 array ] P^3= [ array ll 1 & 3 0 & 1 array ] . . P⁵⁰= [ array cc 1 & 50 0 & 1 array ] So, A^T X⁵⁰ A=P⁵⁰= [ array cc 1 & 50 0 & 1 array ]

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