Question

If `A=[[i,0],[0,i ]]` , n ∈ N, then A⁴ⁿ equals

Options

• `[[0,I],[I,0]]`

• `[[0,0],[0,0]]`

• `[[1,0],[0,1]]`

• `[[0,I],[I,0]]`

Answer 1

`[[1       0],[0      1]]`

\[Here, \] 

\[A = \begin{bmatrix}i & 0 \\ 0 & i\end{bmatrix}\] 

\[ \Rightarrow A^2 = \begin{bmatrix}i & 0 \\ 0 & i\end{bmatrix}\begin{bmatrix}i & 0 \\ 0 & i\end{bmatrix} = \begin{bmatrix}i^2 & 0 \\ 0 & i^2\end{bmatrix}\] 

\[ \Rightarrow A^3 = A^2 . A = \begin{bmatrix}i^2 & 0 \\ 0 & i^2\end{bmatrix}\begin{bmatrix}i & 0 \\ 0 & i\end{bmatrix} = \begin{bmatrix}- i & 0 \\ 0 & - i\end{bmatrix}\] 

\[ \Rightarrow A^4 = A^3 . A = \begin{bmatrix}- i & 0 \\ 0 & - i\end{bmatrix}\begin{bmatrix}i & 0 \\ 0 & i\end{bmatrix} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\] 

So,

\[\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\] is repeated on multiple of 4 and 4n is a multiple of 4.

Thus ,

`A^(4n)=[[1,0],[0,1]]`