Let's break down the given matrices and determine the results step by step as required.
### 1. Calculate the size of the product [tex]\( C = AB \)[/tex]
Given:
[tex]\[
A = \begin{pmatrix}
1 & -1 \\
0 & 3
\end{pmatrix}, \quad
B = \begin{pmatrix}
0 & 2 \\
1 & -1
\end{pmatrix}
\][/tex]
First, let's determine the size of the resulting matrix [tex]\( C \)[/tex]:
- Matrix [tex]\( A \)[/tex] is [tex]\( 2 \times 2 \)[/tex]
- Matrix [tex]\( B \)[/tex] is [tex]\( 2 \times 2 \)[/tex]
When multiplying two matrices, the resulting matrix has the number of rows of the first matrix and the number of columns of the second matrix. Thus, the product [tex]\( C = AB \)[/tex] will also be of size [tex]\( 2 \times 2 \)[/tex].
Therefore, the size of the product [tex]\( C \)[/tex] is:
[tex]\[ \boxed{(2, 2)} \][/tex]
### 2. Calculate the product [tex]\( D = BA \)[/tex]
Let's find the elements of the product [tex]\( D = BA \)[/tex]:
[tex]\[
B = \begin{pmatrix}
0 & 2 \\
1 & -1
\end{pmatrix}, \quad
A = \begin{pmatrix}
1 & -1 \\
0 & 3
\end{pmatrix}
\][/tex]
Now, we will compute each element of [tex]\( D \)[/tex]:
[tex]\[
d_{11} = (0 \cdot 1) + (2 \cdot 0) = 0
\][/tex]
[tex]\[
d_{12} = (0 \cdot -1) + (2 \cdot 3) = 6
\][/tex]
[tex]\[
d_{21} = (1 \cdot 1) + (-1 \cdot 0) = 1
\][/tex]
[tex]\[
d_{22} = (1 \cdot -1) + (-1 \cdot 3) = -4
\][/tex]
Therefore, the elements of [tex]\( D \)[/tex] are:
[tex]\[
\begin{aligned}
d_{11} &= \boxed{0} \\
d_{12} &= \boxed{6} \\
d_{21} &= \boxed{1} \\
d_{22} &= \boxed{-4}
\end{aligned}
\][/tex]
### 3. Confirming the completion
Finally, the detailed elements of the required solution are:
- The size of the product [tex]\( C = AB \)[/tex] is [tex]\( \boxed{(2, 2)} \)[/tex].
- The elements of the product [tex]\( D \)[/tex] are [tex]\( d_{11} = \boxed{0} \)[/tex], [tex]\( d_{12} = \boxed{6} \)[/tex], [tex]\( d_{21} = \boxed{1} \)[/tex], and [tex]\( d_{22} = \boxed{-4} \)[/tex].
Hence, the values have been correctly identified according to the given tasks.
