1. **Stating the problem:** We are given the matrix $$A = \begin{bmatrix} 2 & 1 \\ 0 & 3 \end{bmatrix}$$ and need to find $$A^3$$, which means multiplying matrix $$A$$ by itself three times: $$A^3 = A \times A \times A$$. 2. **Formula and rules:** Matrix multiplication is associative but not commutative. To find $$A^3$$, we first find $$A^2 = A \times A$$, then multiply the result by $$A$$ again. 3. **Calculate $$A^2$$:** $$ A^2 = \begin{bmatrix} 2 & 1 \\ 0 & 3 \end{bmatrix} \times \begin{bmatrix} 2 & 1 \\ 0 & 3 \end{bmatrix} = \begin{bmatrix} (2)(2) + (1)(0) & (2)(1) + (1)(3) \\ (0)(2) + (3)(0) & (0)(1) + (3)(3) \end{bmatrix} = \begin{bmatrix} 4 & 5 \\ 0 & 9 \end{bmatrix} $$ 4. **Calculate $$A^3 = A^2 \times A$$:** $$ A^3 = \begin{bmatrix} 4 & 5 \\ 0 & 9 \end{bmatrix} \times \begin{bmatrix} 2 & 1 \\ 0 & 3 \end{bmatrix} = \begin{bmatrix} (4)(2) + (5)(0) & (4)(1) + (5)(3) \\ (0)(2) + (9)(0) & (0)(1) + (9)(3) \end{bmatrix} = \begin{bmatrix} 8 & 19 \\ 0 & 27 \end{bmatrix} $$ 5. **Final answer:** $$ A^3 = \begin{bmatrix} 8 & 19 \\ 0 & 27 \end{bmatrix} $$ This is the cube of matrix $$A$$.