Let's solve the problem step-by-step:
1. Multiply the corresponding elements in each column of the given matrix:
The given matrix is:
[tex]\[
\left[
\begin{array}{ccccccc}
48 & 73 & 114 & 30 & 136 & 78 & 35 \\
21 & 32 & 50 & 13 & 58 & 35 & 15
\end{array}
\right]
\][/tex]
By multiplying the elements of each column:
- First column: [tex]\(48 \times 21 = 1008\)[/tex]
- Second column: [tex]\(73 \times 32 = 2336\)[/tex]
- Third column: [tex]\(114 \times 50 = 5700\)[/tex]
- Fourth column: [tex]\(30 \times 13 = 390\)[/tex]
- Fifth column: [tex]\(136 \times 58 = 7888\)[/tex]
- Sixth column: [tex]\(78 \times 35 = 2730\)[/tex]
- Seventh column: [tex]\(35 \times 15 = 525\)[/tex]
Therefore, the resulting values after multiplying corresponding elements are:
[tex]\[
[1008, 2336, 5700, 390, 7888, 2730, 525]
\][/tex]
2. Map the product values to characters:
We map the product values to characters using the rule where [tex]\(A=1, B=2, ..., Z=26\)[/tex]. In problems like these, we usually use modulo 27 to get the values corresponding to letters (including a space as modulo result 0).
- For [tex]\(1008\)[/tex]:
[tex]\[
1008 \mod 27 = 9 \quad \Rightarrow \quad \text{'I'}
\][/tex]
- For [tex]\(2336\)[/tex]:
[tex]\[
2336 \mod 27 = 14 \quad \Rightarrow \quad \text{'N'}
\][/tex]
- For [tex]\(5700\)[/tex]:
[tex]\[
5700 \mod 27 = 12 \quad \Rightarrow \quad \text{'L'}
\][/tex]
- For [tex]\(390\)[/tex]:
[tex]\[
390 \mod 27 = 12 \quad \Rightarrow \quad \text{'L'}
\][/tex]
- For [tex]\(7888\)[/tex]:
[tex]\[
7888 \mod 27 = 26 \quad \Rightarrow \quad \text{'Z'}
\][/tex]
- For [tex]\(2730\)[/tex]:
[tex]\[
2730 \mod 27 = 26 \quad \Rightarrow \quad \text{'Z'}
\][/tex]
- For [tex]\(525\)[/tex]:
[tex]\[
525 \mod 27 = 21 \quad \Rightarrow \quad \text{'U'}
\][/tex]
Therefore, mapping the product values yields the sequence:
[tex]\[
['I', 'N', 'L', 'L', 'Z', 'Z', 'U']
\][/tex]
3. Determine the location of spaces:
Given that the decoded values are:
[tex]\[
INLLZZU
\][/tex]
We need to place spaces accordingly. Considering common English phrases and the given options; the most appropriate match is:
- "FINE IS THE PIANO" fits the given sequence best if broken down logically into a readable phrase that matches the given letter sequence best.
Therefore, the encoded message is: "INCLDCL" or "FINE IS THE PIANO".
Thus, the decoded message after placing spaces appropriately is:
[tex]\[
\boxed{\text{FIND THE INCL}} = {\text{FIND THE PRODUCT}}
\][/tex]
