Mathematics - Matrices Question with Solution | TestHub

Since $A{A}^T=I$ therefore matrix $A$ is orthogonal matrix. Now, $A^T{X}^{50}A=A^T{X}^{49}\left(AP{A}^T\right)A$ $=A^T{X}^{49}AP\left(A^{T}A\right)=A^T{X}^{49}AP$ $=A^T{X}^{48}\left(AP{A}^T\right)AP=A^T{X}^{48}A{P}^2\dots \dots$. $=A^{T}A{P}^{50}=I{P}^{50}=P^{50}$ $\because P=\left[\begin{array}{ll}1& 1\\ 0& 1\end{array}\right]\Rightarrow P^2=\left[\begin{array}{ll}1& 2\\ 0& 1\end{array}\right]$ $\Rightarrow P^3=\left[\begin{array}{ll}1& 3\\ 0& 1\end{array}\right]\dots .$. $\Rightarrow P^{50}=\left[\begin{array}{cc}1& 50\\ 0& 1\end{array}\right]$ So, $A^T{X}^{50}A=P^{50}=\left[\begin{array}{cc}1& 50\\ 0& 1\end{array}\right]$