Question

If \[A = \begin{bmatrix}1 & 1 \\ 1 & 1\end{bmatrix}\] satisfies A⁴ = λA, then write the value of λ.

 

 

Answer 1

\[A = \begin{bmatrix}1 & 1 \\ 1 & 1\end{bmatrix}\]

`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`