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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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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).

edited

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