Cayley- Hamilton Theorem is very useful to find:

- Inverse, of the given matrix
- The higher power of the given matrix

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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$

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$

edited
Nov 29, 2019
by Lakshman Patel RJIT

Cayley- Hamilton Theorem is very useful to find:

- Inverse, of the given matrix
- The higher power of the given matrix

edited
Jan 28, 2019
by Lakshman Patel RJIT

This is an upper triangular matrix, therefore its eigen values or characteristic roots will be all the diagonal elements (1, -1, i, -i)

therefore, (λ – 1)(λ + 1)(λ – i)(λ + i) = 0 will hold true

which will give characteristic equation as:

λ^4 – 1 = 0

and according to Cayley Hamilton theorem every matrix matrix satisfies its own characteristic equation

=> A^4 – I = 0

**A^4 = I**

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