If $A = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}$, then find $A^3$.

$\begin{bmatrix} 1 & 0 \\ 3 & 1 \end{bmatrix}$

The correct answer is Option (3) → $\begin{bmatrix} 1 & 0 \\ 3 & 1 \end{bmatrix}$ ##

Since, $A \cdot A = A^2 = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}$

$= \begin{bmatrix} 1 \times 1 + 0 \times 1 & 1 \times 0 + 0 \times 1 \\ 1 \times 1 + 1 \times 1 & 1 \times 0 + 1 \times 1 \end{bmatrix}$

$= \begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix}$

$A^2 \cdot A = A^3 = \begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}$

$= \begin{bmatrix} 1 \times 1 + 0 \times 1 & 1 \times 0 + 0 \times 1 \\ 2 \times 1 + 1 \times 1 & 2 \times 0 + 1 \times 1 \end{bmatrix}$

$= \begin{bmatrix} 1 & 0 \\ 3 & 1 \end{bmatrix}$