Question
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 X^ 50 A=
BITSAT
Mathematics
Matrices
2024
BITSAT 2024 (Memory Based Paper 1)
BITSAT Mathematics Question (2024) — Solution
Options
- A. [ array ll 0 & 1 \\ 1 & 0 array ]
- B. [ array cc 2 & 1 \\ 0 & -1 array ]
- C. [ array cc 25 & 1 \\ 1 & -25 array ]
- D. [ array cc 1 & 50 \\ 0 & 1 array ]
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 X^ 50 A=A^T X^ 49 (A P A^T ) A =A^T X^ 49 A P (A^T A )=A^T X^ 49 A P =A^T X^ 48 (A P A^T ) A P=A^T X^ 48 A P^2 . =A^T A P^ 50 =I P^ 50 =P^ 50 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 ] .. P^ 50 = [ array cc 1 & 50 \\ 0 & 1 array ] So, A^T X^ 50 A=P^ 50 = [ array cc 1 & 50 \\ 0 & 1 array ]
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