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Let A be the $ 2 × 2 $ matrix with elements $a_{11} = a_{12} = a_{21} = +1 $ and $ a_{22} = −1 $ . Then the eigenvalues of the matrix $A^{19}$ are

  1. $1024$ and $−1024$
  2. $1024\sqrt{2}$ and $−1024 \sqrt{2}$
  3. $4 \sqrt{2}$ and $−4 \sqrt{2}$
  4. $512 \sqrt{2}$ and $−512 \sqrt{2}$
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Its based on the property of eigen values . If A is a matrix and has eigen values x1,x2,,,,,,,xn then eigen values of A^n will x1^n,x2^n......xn^n

So for given question eigen value of A is 1.414 so we have to find (1.414)^19 which gives option d.

eNJOY~!
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