+12 votes
646 views
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 ____
asked
retagged | 646 views
+2

This might help ....

## 1 Answer

+27 votes
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$
answered by Boss (31.3k points)
edited
0
Cayley Hamilton theorem is very useful to find (1) Inverse of given matrix

(2) Higher power of given matrix

+3 votes
2 answers
1
+13 votes
2 answers
2