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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 $−512 \sqrt{2}$
asked in Linear Algebra by Boss (18k points)
edited by | 1.6k views

2 Answers

+27 votes
Best answer

Characteristic Equation is $ A  -\mu I = 0$

$ \begin{vmatrix} 1-\mu & 1\\1 &-1-\mu\end{vmatrix} = 0$

$\implies (1-\mu)(-1-\mu)-1=0$

$\implies -1-\mu+\mu+\mu^{2}-1=0$

$\implies \mu^{2}-2=0$

$\implies \mu=+\sqrt 2 $ and $-\sqrt 2$

so according to properties of Eigen values,

eigen values of $A^{19}$=(eigen value of A)$^{19}$

=$(\sqrt 2)^{19}$ and $(-\sqrt 2)^{19}$

 Hence Ans is option (D).

answered by (421 points)
edited by
+4 votes

Eigenvalues Of Matrix Powers

Suppose A is a square matrix,λ is an eigenvalue of A, and s≥0 is an integer. Then λs is an eigenvalue of As.

http://linear.ups.edu/html/section-PEE.html

answered by Loyal (7.6k points)


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