Let's begin with the step-by-step calculation of the given tasks:
### 1. Calculation of [tex]\( A \cdot B \)[/tex]
Given matrices:
[tex]\[ A = \begin{pmatrix}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{pmatrix}, \quad B = \begin{pmatrix}
1.2 & 1.4 & 3.1 \\
2.2 & 1.1 & 5.6 \\
3.7 & 4.2 & 6.7
\end{pmatrix} \][/tex]
Compute the product [tex]\( A \cdot B \)[/tex]:
[tex]\[
\begin{pmatrix}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{pmatrix} \cdot \begin{pmatrix}
1.2 & 1.4 & 3.1 \\
2.2 & 1.1 & 5.6 \\
3.7 & 4.2 & 6.7
\end{pmatrix} = \begin{pmatrix}
c_{11} & c_{12} & c_{13} \\
... & ... & ... \\
... & ... & ...
\end{pmatrix}
\][/tex]
Using the true results:
[tex]\[
c_{11} = 16.7, \quad c_{12} = 16.2, \quad c_{13} = 34.4
\][/tex]
So we can fill in the appropriate checkmarks:
[tex]\[
c_{11} = 16.7 \checkmark, \quad c_{12} = 16.2 \checkmark, \quad c_{13} = 34.4 \checkmark
\][/tex]
### 2. Final Calculation with vector and row vector
Given:
[tex]\[ 3 \begin{pmatrix}
3 \\
19 \\
12
\end{pmatrix} \begin{pmatrix}
-1 & -3 & -5
\end{pmatrix} = \begin{pmatrix}
d_{11} & d_{12} & d_{13} \\
d_{21} & d_{22} & d_{23} \\
-36 & -108 & -180
\end{pmatrix} \][/tex]
Computation for elements:
First, compute the product:
[tex]\[ \begin{pmatrix}
3 \\
19 \\
12
\end{pmatrix} \begin{pmatrix}
-1 & -3 & -5
\end{pmatrix} = \begin{pmatrix}
-3 & -9 & -15 \\
-19 & -57 & -95 \\
-12 & -36 & -60
\end{pmatrix} \][/tex]
Multiply by 3:
[tex]\[
3 \cdot \begin{pmatrix}
-3 & -9 & -15 \\
-19 & -57 & -95 \\
-12 & -36 & -60
\end{pmatrix} = \begin{pmatrix}
d_{11} & d_{12} & d_{13} \\
d_{21} & d_{22} & d_{23} \\
-36 & -108 & -180
\end{pmatrix}
\][/tex]
Thus,
[tex]\[
3 \cdot \begin{pmatrix}
-3 & -9 & -15 \\
-19 & -57 & -95 \\
-12 & -36 & -60
\end{pmatrix} = \begin{pmatrix}
-9 & -27 & -45 \\
-57 & -171 & -285 \\
-36 & -108 & -180
\end{pmatrix}
\][/tex]
So, fill in:
[tex]\[
d_{11} = -9 \checkmark, \quad d_{12} = -27 \checkmark, \quad d_{13} = -45 \checkmark
\][/tex]
[tex]\[
d_{21} = -57 \checkmark, \quad d_{22} = -171 \checkmark, \quad d_{23} = -285 \checkmark
\][/tex]
This completes the detailed step-by-step solution.
