Question-95

If \(\displaystyle A=\begin{bmatrix} 1 & 2\\ 2 & -1 \end{bmatrix}\), which of the following is \(\displaystyle A^{8}\)?

• \(\displaystyle \begin{bmatrix} 25 & 0\\ 0 & 25 \end{bmatrix}\)

• \(\displaystyle \begin{bmatrix} 125 & 0\\ 0 & 125 \end{bmatrix}\)

• \(\displaystyle \begin{bmatrix} 625 & 0\\ 0 & 625 \end{bmatrix}\)

• \(\displaystyle \begin{bmatrix} 3125 & 0\\ 0 & 3125 \end{bmatrix}\)

NoteHint

Find \(A^2\).

NoteAnswer

• \(\displaystyle \begin{bmatrix} 25 & 0\\ 0 & 25 \end{bmatrix}\)

• \(\displaystyle \begin{bmatrix} 125 & 0\\ 0 & 125 \end{bmatrix}\)

• \(\displaystyle \begin{bmatrix} 625 & 0\\ 0 & 625 \end{bmatrix}\)

• \(\displaystyle \begin{bmatrix} 3125 & 0\\ 0 & 3125 \end{bmatrix}\)

NoteSolution

Let us first compute \(\displaystyle A^{2}\):

\[ \begin{aligned} A^{2} & =\begin{bmatrix} 1 & 2\\ 2 & -1 \end{bmatrix}\begin{bmatrix} 1 & 2\\ 2 & -1 \end{bmatrix}\\ & =\begin{bmatrix} 5 & 0\\ 0 & 5 \end{bmatrix}\\ & =5I \end{aligned} \]

Therefore, \(\displaystyle A^{8} =5^{4} I=625I\).