Question

If \[A = \begin{bmatrix}2 & - 1 \\ 3 & - 2\end{bmatrix},\text{ then } A^n =\] ______________ .

Options

• \[A^n = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\], if n is an even natural number

• \[A^n = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\] , if n is an odd natural number

• \[A^n = \begin{bmatrix}- 1 & 0 \\ 0 & 1\end{bmatrix}\], if n ∈ N

• none of these

Answer 1

\[A = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\], if n is an even natural number
\[A = \begin{bmatrix}2 & - 1 \\ 3 & - 2\end{bmatrix}\]
\[ A^2 = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\]
\[ \Rightarrow A \times A = I\]
\[ \Rightarrow A^{- 1} = A\]
Generally,
\[A^n = (A A^{- 1} )^{n/2}\text{ when n is even .} \]
\[ A^n = A(A A^{- 1} )^{n/2} = A\text{ when n is odd }. \]
\[\text{Thus, } A^n = I \text{ when n is even }.\]