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If $A =\begin{bmatrix} 2 &i \\ i & 0 \end{bmatrix}$ , the trace of $A^{10}$ is

  1. $2$
  2. $2(1+i)$
  3. $0$
  4. $2^{10}$
asked in Linear Algebra by Boss (41k points) | 48 views

1 Answer

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By matrix multiplication , $A^2$ = $\begin{bmatrix} 3 &2i \\ 2i & -1 \end{bmatrix}$ whose trace is 2.

$A^3$ = $\begin{bmatrix} 4 & 3i\\ 3i & -2 \end{bmatrix}$ whose trace is 2.
So we see the trace of the original matrix remains the same irrespective of repeated multiplication with itself. Therefore trace of $A^{10}$ = 2
answered by Loyal (7k points)

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