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Question
If \[A = \begin{bmatrix}1 & 1 \\ 1 & 1\end{bmatrix}\] satisfies A4 = λA, then write the value of λ.
Solution
`A^2=A.A`
`⇒ A^2=[[1,1],[1,1]]` `[[1,1],[1,1]]`
`⇒ A^2=[[1+1,1+1],[1+1,1+1]]`
`⇒ A^2=[[2,2],[2,2]]`
Now
`A^2=A^2 A^2`
`⇒A^4=[[2,2],[2,2]]` `[[2,2],[2,2]]`
`⇒A^4=[[4+4,4+4],[4+4,4+4]]`
`⇒A^4=[[8,8],[8,8]]`
Also ,
`A^4=λ A`
`⇒ [[8,8],[8,8]]=λ[[1,1],[1,1]]`
`⇒ [[8,8],[8,8]]=[[λ,λ],[λ,λ]]`
`⇒ λ=8`
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