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Sagot :
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]
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]
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