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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 ....

## 1 Answer

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Best answer
Let λ be eighen value

Characteristic polynomial is

$(1-λ)(-1-λ)(i-λ)(-i-λ)$

$=\left ( \lambda ^{2}-1 \right )\left ( \lambda ^{2}+1 \right )$

$=\lambda ^{4}-1$

Characteristic equation is $\lambda ^{4}-1=0$

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

So, $A^{4}=$$I$
by Boss (30.9k points)
edited
+1
Cayley Hamilton theorem is very useful to find (1) Inverse of given matrix

(2) Higher power of given matrix
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from the characteristic equation how we can conclude A^4=I ?

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see the above videos.

According to Cayley Hamilton theorem, every matrix satisfies its own characteristic equation and vice versa.

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got it..thanks!

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