If A = (\begin{bmatrix} 0 & i \\ –i & 0 \end{bmatrix}) then... | MATH - TYPES OF MATRICES, ALGEBRA OF MATRICES | SolveeIt

If A = $\begin{bmatrix} 0 & i \\ –i & 0 \end{bmatrix}$ then A⁴⁰ is equal to

$\begin{bmatrix} 0 & 40i \\ –40i & 0 \end{bmatrix}$

$\begin{bmatrix} 0 & 20i \\ –20i & 0 \end{bmatrix}$

$\begin{bmatrix} 20 & 0 \\ 0 & 20 \end{bmatrix}$

$\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

Answer

$\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

Explanation

A = $\begin{bmatrix} 0 & i \\ –i & 0 \end{bmatrix}$

A² = $\begin{bmatrix} 0 & i \\ –i & 0 \end{bmatrix}$ $\overset{\downarrow}{\begin{bmatrix} 0 & i \\ –i & 0 \end{bmatrix}}$

A² = $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ = I

A⁴⁰ = (A²)²⁰ = I²⁰

A⁴⁰ = I

Question: If A = \(\begin{bmatrix} 0 & i \\ –i & 0 \end{bmatrix}\) then A<sup>40</sup> is equal to...