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Sagot :
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]
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]
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