If A = {bmatrix}1&0\\0&-1{bmatrix} , P = {bmatrix}1&1\\0&1{bmatrix} and X = APA^T then A^T X^{50} A =

Correct Answer is : [15001]{bmatrix}1&50\\0&1{bmatrix}[10501]

Correct Answer is : [15001]{bmatrix}1&50\\0&1{bmatrix}[10501]

Test Index

AP EAMCET Engineering 20 Apr 2019 Shift 1 Paper

Section: Mathematics

Question : 4 of 160

Marks: +1, -0

If A =

\begin{bmatrix}1&0\\0&-1\end{bmatrix}

\begin{bmatrix}1&1\\0&1\end{bmatrix}

APA^T

A^T X^{50} A

Solution:

Consider the matrix,

A=\begin{bmatrix}1&0\\0&-1\end{bmatrix}

since above matrix satisfies

A A^{T}=I

so, it is an orthogonal matrix then,

A^{T} X^{50} A = A^{T} X^{49} (APA^{T}) A

= A^{T} X^{49} AP (A^{T} A)

= A^{T} X^{49} AP

= A^{T} X^{48} AP^{2} \dots

Further simplify the above,

A^{T} X^{50} A = A^{T} A P^{50}

= I P^{50}

= P^{50}

This implies,

P=\begin{bmatrix}1&1\\0&1\end{bmatrix}

P^{2} =\begin{bmatrix}1&2\\0&1\end{bmatrix}

P^{3} =\begin{bmatrix}1&3\\0&1\end{bmatrix}

\vdots

P^{50} =\begin{bmatrix}1&50\\0&1\end{bmatrix}

Therefore, the correct option is (4)

Go to Question:

Prev Question

Next Question