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Let A be a 4X4 matrix such that sum of elements in each row is1.If sum of all the elements in A^10 is n,then find number of solutions of the equation sin^-1 x+cos^-1 x^2=2pi/n.

Let A be a 4X4 matrix such that sum of elements in each row is1.If sum of all the elements in A^10 is n,then find number of solutions of the equation sin^-1 x+cos^-1 x^2=2pi/n. 

Grade:12

1 Answers

mycroft holmes
272 Points
7 years ago
Note that the sum of elements in the row of matrix A can be obtained by the operation
A \times \begin{bmatrix} 1\\ 1\\ 1\\ 1 \end{bmatrix}
We are now given that A \times \begin{bmatrix} 1\\ 1\\ 1\\ 1 \end{bmatrix} = \begin{bmatrix} 1\\ 1\\ 1\\ 1 \end{bmatrix}
 
So we also get
 
A^2 \times \begin{bmatrix} 1\\ 1\\ 1\\ 1 \end{bmatrix} = A \times \left (A \times \begin{bmatrix} 1\\ 1\\ 1\\ 1 \end{bmatrix} \right)
 
= A \times \begin{bmatrix} 1\\ 1\\ 1\\ 1 \end{bmatrix} = \begin{bmatrix} 1\\ 1\\ 1\\ 1 \end{bmatrix}
 
So, the result is the same when we use A10 for the multiplication. So the sum of elements in each row of A10 is 1. Hence the sum of all elements is simply 4.
 
So the equation to be solved is \sin^{-1} x + \cos^{-1}x^2 = \frac{\pi}{2} \Rightarrow \cos^{-1}x^2 = \frac{\pi}{2}-\sin^{-1} x =\cos^{-1}x
 
Since cos-1x is injective, we must have x = x2 which has solutions x=0,1

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