Let's solve the problem by performing the matrix additions and then finding the values of [tex]\( n \)[/tex], [tex]\( m \)[/tex], and [tex]\( s \)[/tex] before checking which given statements are true.
1. Matrix Addition Calculation:
First, add the second and third matrices:
[tex]\[
\left[\begin{array}{ccc}
-5 & 0 & 4 \\
0 & -12 & 2 \\
1 & -18 & 8
\end{array}\right]
+
\left[\begin{array}{ccc}
1 & -6 & -7 \\
4 & 0 & 3 \\
14 & -11 & 5
\end{array}\right]
=
\left[\begin{array}{ccc}
-4 & -6 & -3 \\
4 & -12 & 5 \\
15 & -29 & 13
\end{array}\right]
\][/tex]
Next, add the resulting matrix to the first matrix:
[tex]\[
\left[\begin{array}{ccc}
3 & -1 & n \\
6 & m-1 & -9 \\
17 & s+1 & 7
\end{array}\right]
+
\left[\begin{array}{ccc}
-4 & -6 & -3 \\
4 & -12 & 5 \\
15 & -29 & 13
\end{array}\right]
=
\left[\begin{array}{ccc}
-1 & -7 & n-3 \\
10 & m-13 & -4 \\
32 & s-28 & 20
\end{array}\right]
\][/tex]
Now compare this to the resulting matrix given in the problem:
[tex]\[
\left[\begin{array}{ccc}
-1 & -7 & 18 \\
10 & 12 & -4 \\
32 & -28 & 20
\end{array}\right]
\][/tex]
From this, we find:
[tex]\[
n - 3 = 18 \quad \Rightarrow \quad n = 21
\][/tex]
[tex]\[
m - 13 = 12 \quad \Rightarrow \quad m = 25
\][/tex]
[tex]\[
s - 28 = -28 \quad \Rightarrow \quad s = 0
\][/tex]
2. Checking the Statements:
Let's check each given statement with the values we just found:
1. [tex]\((n + 4) - 7 = 18\)[/tex]:
[tex]\[
(21 + 4) - 7 = 25 - 7 = 18 \quad \text{(True)}
\][/tex]
2. [tex]\(((m - 1) + 12) + 0 = 12\)[/tex]:
[tex]\[
((25 - 1) + 12) + 0 = (24 + 12) + 0 = 36 \quad \text{(False)}
\][/tex]
3. [tex]\(((s + 1) - 18) + 11 = -28\)[/tex]:
[tex]\[
((0 + 1) - 18) + 11 = (1 - 18) + 11 = -17 + 11 = -6 \quad \text{(False)}
\][/tex]
4. [tex]\(n + (m - 1) + (s + 1) = -28\)[/tex]:
[tex]\[
21 + (25 - 1) + (0 + 1) = 21 + 24 + 1 = 46 \quad \text{(False)}
\][/tex]
Hence, the only true statement is:
[tex]\[
(n + 4) - 7 = 18
\][/tex]
