Let's calculate the product of the given matrices step-by-step.
We need to multiply the matrix:
[tex]\[
A = \begin{pmatrix}
4 & 3 \\
-1 & -1
\end{pmatrix}
\][/tex]
by the matrix:
[tex]\[
B = \begin{pmatrix}
-3 & 1 \\
-2 & 2
\end{pmatrix}
\][/tex]
The result of the multiplication will be a new matrix [tex]\(C\)[/tex] of the form:
[tex]\[
C = A \times B = \begin{pmatrix}
c_{11} & c_{12} \\
c_{21} & c_{22}
\end{pmatrix}
\][/tex]
Where each element [tex]\(c_{ij}\)[/tex] in the resulting matrix [tex]\(C\)[/tex] is calculated by taking the dot product of the [tex]\(i\)[/tex]-th row of matrix [tex]\(A\)[/tex] with the [tex]\(j\)[/tex]-th column of matrix [tex]\(B\)[/tex]:
1. Calculate [tex]\(c_{11}\)[/tex]:
[tex]\[
c_{11} = 4 \cdot (-3) + 3 \cdot (-2)
\][/tex]
[tex]\[
c_{11} = -12 + (-6)
\][/tex]
[tex]\[
c_{11} = -18
\][/tex]
2. Calculate [tex]\(c_{12}\)[/tex]:
[tex]\[
c_{12} = 4 \cdot 1 + 3 \cdot 2
\][/tex]
[tex]\[
c_{12} = 4 + 6
\][/tex]
[tex]\[
c_{12} = 10
\][/tex]
3. Calculate [tex]\(c_{21}\)[/tex]:
[tex]\[
c_{21} = -1 \cdot (-3) + (-1) \cdot (-2)
\][/tex]
[tex]\[
c_{21} = 3 + 2
\][/tex]
[tex]\[
c_{21} = 5
\][/tex]
4. Calculate [tex]\(c_{22}\)[/tex]:
[tex]\[
c_{22} = -1 \cdot 1 + (-1) \cdot 2
\][/tex]
[tex]\[
c_{22} = -1 + (-2)
\][/tex]
[tex]\[
c_{22} = -3
\][/tex]
So, the resulting matrix [tex]\(C\)[/tex] is:
[tex]\[
C = \begin{pmatrix}
-18 & 10 \\
5 & -3
\end{pmatrix}
\][/tex]
Thus, the product of the two matrices is:
[tex]\[
\boxed{\begin{pmatrix}
-18 & 10 \\
5 & -3
\end{pmatrix}}
\][/tex]
