The Gateway to Computer Science Excellence
First time here? Checkout the FAQ!
x
+13 votes
863 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 in Linear Algebra by Veteran (59.7k points)
retagged by | 863 views
+5

This might help ....

1 Answer

+30 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 (31.9k points)
edited by
+1
Cayley Hamilton theorem is very useful to find (1) Inverse of given matrix

                                                                           (2) Higher power of given matrix
0

@Lakshman Patel RJIT

from the characteristic equation how we can conclude A^4=I ?
 

0

@himgta

see the above videos.

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

0
got it..thanks!

Related questions



Quick search syntax
tags tag:apple
author user:martin
title title:apple
content content:apple
exclude -tag:apple
force match +apple
views views:100
score score:10
answers answers:2
is accepted isaccepted:true
is closed isclosed:true

44,490 questions
49,940 answers
165,704 comments
65,910 users