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= )

Step-by-step solution

Given matrix (A= [ array cc 1 & 0 0 & -1 array ] ) is orthogonal matrix, because (A A^T=I ). ( aligned & So, 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 ] aligned ) So, ( A^T X⁵⁰ A=P⁵⁰= [ array cc 1 & 50 0 & 1 array ] ) Hence, option (4) is correct.