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If \(A=\begin{bmatrix}i & 0\\ 0 & i\end{bmatrix}\), which of the following is NOT equal to \(A^7\)?

1. \(\begin{bmatrix}0 & i\\ i & 0\end{bmatrix}\)
2. \(\begin{bmatrix}1 & i\\ i & 1\end{bmatrix}\)
3. \(\begin{bmatrix}0 & 1\\ 1 & 0\end{bmatrix}\)
4. None of these

Correct answer: \(\begin{bmatrix}1 & i\\ i & 1\end{bmatrix}\)

Solution

Here \(A=iI\), so \(A^7=i^7I\). Since \(i^4=1\), we get \(i^7=i^4\cdot i^3=1\cdot(-i)=-i\), hence \(A^7=-iI\). The only option that is clearly not equal to a diagonal matrix of this form is \(\begin{bmatrix}1&i\\ i&1\end{bmatrix}\).

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