Find \(A^2\) and \(A^3\)
Question:
If \[ A=\begin{bmatrix}1 & 1 \\ 0 & 1\end{bmatrix} \] show that: \[ A^2=\begin{bmatrix}1 & 2 \\ 0 & 1\end{bmatrix}, \quad A^3=\begin{bmatrix}1 & 3 \\ 0 & 1\end{bmatrix} \]
If \[ A=\begin{bmatrix}1 & 1 \\ 0 & 1\end{bmatrix} \] show that: \[ A^2=\begin{bmatrix}1 & 2 \\ 0 & 1\end{bmatrix}, \quad A^3=\begin{bmatrix}1 & 3 \\ 0 & 1\end{bmatrix} \]
Solution:
Step 1: Compute \(A^2 = A \cdot A\)
\[ A^2 = \begin{bmatrix}1 & 1 \\ 0 & 1\end{bmatrix} \begin{bmatrix}1 & 1 \\ 0 & 1\end{bmatrix} \] \[ = \begin{bmatrix} 1(1)+1(0) & 1(1)+1(1) \\ 0(1)+1(0) & 0(1)+1(1) \end{bmatrix} = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix} \]Step 2: Compute \(A^3 = A^2 \cdot A\)
\[ A^3 = \begin{bmatrix}1 & 2 \\ 0 & 1\end{bmatrix} \begin{bmatrix}1 & 1 \\ 0 & 1\end{bmatrix} \] \[ = \begin{bmatrix} 1(1)+2(0) & 1(1)+2(1) \\ 0(1)+1(0) & 0(1)+1(1) \end{bmatrix} = \begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix} \]Conclusion:
\[ A^2=\begin{bmatrix}1 & 2 \\ 0 & 1\end{bmatrix}, \quad A^3=\begin{bmatrix}1 & 3 \\ 0 & 1\end{bmatrix} \]Hence proved.