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If $A = \begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & -1 & 0 & -1 \\ 0 & 0 & i & i \\ 0 & 0 & 0 & -i \end{pmatrix}$ the matrix $A^4$, calculated by the use of Cayley-Hamilton theorem or otherwise, is ____
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This might help ....

Let λ be eighen value

Characteristic polynomial is

(1-λ)(-1-λ)(i-λ)(-i-λ)

=(λ2-1)(λ2+1)

4-1

Characteristic equation is λ4-1=0

According to Cayley Hamilton theorem every matrix matrix satisfies its own characteristic equation

So A4=I

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Cayley Hamilton theorem is very useful to find (1) Inverse of given matrix

(2) Higher power of given matrix