Step 1: Write down the given matrix $P$. $$P = \begin{pmatrix} 4 & 3 \\ 1 & 2 \end{pmatrix}$$ Step 2: Calculate $P^2$ by multiplying $P$ by $P$. $$P^2 = P \times P = \begin{pmatrix} 4 & 3 \\ 1 & 2 \end{pmatrix} \begin{pmatrix} 4 & 3 \\ 1 & 2 \end{pmatrix}$$ Step 3: Perform the matrix multiplication. To find the element in row $i$ and column $j$ of the product matrix, multiply the elements of row $i$ of the first matrix by the corresponding elements of column $j$ of the second matrix and sum the products. • Top-left element (row 1, column 1): $$(4 \times 4) + (3 \times 1) = 16 + 3 = 19$$ • Top-right element (row 1, column 2): $$(4 \times 3) + (3 \times 2) = 12 + 6 = 18$$ • Bottom-left element (row 2, column 1): $$(1 \times 4) + (2 \times 1) = 4 + 2 = 6$$ • Bottom-right element (row 2, column 2): $$(1 \times 3) + (2 \times 2) = 3 + 4 = 7$$ Step 4: Construct the resulting matrix $P^2$. $$P^2 = \begin{pmatrix} 19 & 18 \\ 6 & 7 \end{pmatrix}$$ The final answer is $\boxed{\begin{pmatrix} 19 & 18 \\ 6 & 7 \end{pmatrix}}$.