To calculate the product, we need to perform a step-by-step multiplication of matrix [tex]\( A \)[/tex] by 5, and then verify the matrices involved. Let's start by addressing the components provided:
1. The product of 5 with matrix [tex]\( A \)[/tex]:
[tex]\[
5\left[\begin{array}{ccc}
2 & 3 & 4 \\
9 & -1 & -7 \\
11 & 5 & -3
\end{array}\right] = \left[\begin{array}{ccc}
10 & 15 & 20 \\
45 & -5 & -35 \\
55 & 25 & -15
\end{array}\right]
\][/tex]
The operation of multiplying 5 with each element of matrix [tex]\( A \)[/tex] results in:
[tex]\[
\left[\begin{array}{ccc}
10 & 15 & 20 \\
45 & -5 & -35 \\
55 & 25 & -15
\end{array}\right]
\][/tex]
This confirms matrix [tex]\( B \)[/tex] as:
[tex]\[
\left[\begin{array}{ccc}
10 & 15 & 20 \\
45 & -5 & -35 \\
55 & 25 & -15
\end{array}\right]
\][/tex]
2. Given matrices:
- The first matrix given is:
[tex]\[
\left[\begin{array}{ccc}
10 & 15 & 20 \\
45 & -5 & -35 \\
55 & 25 & -15
\end{array}\right]
\][/tex]
Which matches the matrix [tex]\( B \)[/tex].
- The second matrix given is:
[tex]\[
\left[\begin{array}{ccc}
10 & 3 & 4 \\
45 & -1 & -7 \\
55 & 5 & -3
\end{array}\right]
\][/tex]
Which is a different transformation of the elements of matrix [tex]\( A \)[/tex].
- The third matrix given is:
[tex]\[
\left[\begin{array}{ccc}
7 & 8 & 9 \\
14 & 4 & -2
\end{array}\right]
\][/tex]
Taking into account all given information, we summarize the matrices transformations and products as follows:
- The first step is computing 5 times matrix [tex]\( A \)[/tex].
- The second step is identifying the given matrices and comparing them with our computed results.
Final results for all matrix computations are:
[tex]\[
\left[\begin{array}{ccc}
10 & 15 & 20 \\
45 & -5 & -35 \\
55 & 25 & -15
\end{array}\right],
\left[\begin{array}{ccc}
10 & 15 & 20 \\
45 & -5 & -35 \\
55 & 25 & -15
\end{array}\right],
\left[\begin{array}{ccc}
10 & 3 & 4 \\
45 & -1 & -7 \\
55 & 5 & -3
\end{array}\right],
\left[\begin{array}{ccc}
7 & 8 & 9 \\
14 & 4 & -2
\end{array}\right]
\][/tex]
