To find the product of the two matrices, we will multiply the first matrix by the second matrix.
Let \( A \) be the first matrix:
\[ A = \begin{pmatrix} 0 & 4 \ -3 & 0 \end{pmatrix} \]
And let \( B \) be the second matrix:
\[ B = \begin{pmatrix} -1 & -1 \ 0 & -2 \end{pmatrix} \]
To compute the product \( C = A \cdot B \), we'll use the formula for matrix multiplication, where the element \( c_{ij} \) in the resulting matrix \( C \) is calculated as:
\[ c_{ij} = \sum_{k} a_{ik} b_{kj} \]
Calculation of \( C \)
Element \( c_{11} \): \[ c_{11} = a_{11}b_{11} + a_{12}b_{21} = 0 \cdot (-1) + 4 \cdot 0 = 0 \]
Element \( c_{12} \): \[ c_{12} = a_{11}b_{12} + a_{12}b_{22} = 0 \cdot (-1) + 4 \cdot (-2) = 0 - 8 = -8 \]
Element \( c_{21} \): \[ c_{21} = a_{21}b_{11} + a_{22}b_{21} = (-3) \cdot (-1) + 0 \cdot 0 = 3 + 0 = 3 \]
Element \( c_{22} \): \[ c_{22} = a_{21}b_{12} + a_{22}b_{22} = (-3) \cdot (-1) + 0 \cdot (-2) = 3 + 0 = 3 \]
Putting the elements together
Thus, the product matrix \( C \) is:
\[ C = \begin{pmatrix} c_{11} & c_{12} \ c_{21} & c_{22} \end{pmatrix} = \begin{pmatrix} 0 & -8 \ 3 & 3 \end{pmatrix} \]
Hence, the product of the two matrices is:
\[ \boxed{\begin{pmatrix} 0 & -8 \ 3 & 3 \end{pmatrix}} \]