Write down the given matrix P.

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):

$$(1 \times 3) + (2 \times 2) = 3 + 4 = 7$$

$$P^2 = \begin{pmatrix} 19 & 18 \\ 6 & 7 \end{pmatrix}$$

The final answer is $\begin{pmatrix} 19 & 18 \\ 6 & 7 \end{pmatrix}$.