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Sagot :
Let's analyze the given matrices and their resulting matrix operations step-by-step.
The matrices provided are:
[tex]$ A = \left[\begin{array}{rr} 2 & -5 \\ 1 & 0 \end{array}\right], \quad B = \left[\begin{array}{rr} -2 & -5 \\ 1 & 0 \end{array}\right] $[/tex]
The required matrix operations are:
- [tex]$2(A + B)$[/tex]
- [tex]$2A + 3A$[/tex]
- [tex]$2(A - B)$[/tex]
- [tex]$2A - B$[/tex]
- [tex]$2A$[/tex]
And the resulting matrices are:
- [tex]$\left[\begin{array}{rr}6 & -5 \\ 1 & 0\end{array}\right]$[/tex]
- [tex]$\left[\begin{array}{cc}10 & -25 \\ 5 & 0\end{array}\right]$[/tex]
- [tex]$\left[\begin{array}{ll}8 & 0 \\ 0 & 0\end{array}\right]$[/tex]
Let's match each matrix operation to its result.
### Matching Matrix Operations with Results:
1. [tex]$2(A + B)$[/tex]:
Calculate [tex]$A + B$[/tex]:
[tex]$ A + B = \left[\begin{array}{rr} 2 & -5 \\ 1 & 0 \end{array}\right] + \left[\begin{array}{rr} -2 & -5 \\ 1 & 0 \end{array}\right] = \left[\begin{array}{rr} 0 & -10 \\ 2 & 0 \end{array}\right] $[/tex]
Now multiply by 2:
[tex]$ 2(A + B) = 2 \times \left[\begin{array}{rr} 0 & -10 \\ 2 & 0 \end{array}\right] = \left[\begin{array}{rr} 0 & -20 \\ 4 & 0 \end{array}\right] $[/tex]
The result is not among the matrices provided, so this operation doesn’t match any of our given results.
2. [tex]$2A + 3A$[/tex]:
Calculate:
[tex]$ 2A + 3A = 5A = 5 \times \left[\begin{array}{rr} 2 & -5 \\ 1 & 0 \end{array}\right] = \left[\begin{array}{rr} 10 & -25 \\ 5 & 0 \end{array}\right] $[/tex]
Matching this matrix:
[tex]$ \left[\begin{array}{cc}10 & -25 \\ 5 & 0\end{array}\right] \longrightarrow 2A + 3A $[/tex]
3. [tex]$2(A - B)$[/tex]:
Calculate [tex]$A - B$[/tex]:
[tex]$ A - B = \left[\begin{array}{rr} 2 & -5 \\ 1 & 0 \end{array}\right] - \left[\begin{array}{rr} -2 & -5 \\ 1 & 0 \end{array}\right] = \left[\begin{array}{rr} 4 & 0 \\ 0 & 0 \end{array}\right] $[/tex]
Now multiply by 2:
[tex]$ 2(A - B) = 2 \times \left[\begin{array}{rr} 4 & 0 \\ 0 & 0 \end{array}\right] = \left[\begin{array}{rr} 8 & 0 \\ 0 & 0 \end{array}\right] $[/tex]
Matching this matrix:
[tex]$ \left[\begin{array}{ll}8 & 0 \\ 0 & 0\end{array}\right] \longrightarrow 2(A - B) $[/tex]
4. [tex]$2A - B$[/tex]:
Calculate:
[tex]$ 2A - B = 2 \times \left[\begin{array}{rr} 2 & -5 \\ 1 & 0 \end{array}\right] - \left[\begin{array}{rr} -2 & -5 \\ 1 & 0 \end{array}\right] = \left[\begin{array}{rr} 4 & -10 \\ 2 & 0 \end{array}\right] - \left[\begin{array}{rr} -2 & -5 \\ 1 & 0 \end{array}\right] = \left[\begin{array}{rr} 6 & -5 \\ 1 & 0 \end{array}\right] $[/tex]
Matching this matrix:
[tex]$ \left[\begin{array}{rr}6 & -5 \\ 1 & 0\end{array}\right] \longrightarrow 2A - B $[/tex]
5. [tex]$2A$[/tex]:
Calculate:
[tex]$ 2A = 2 \times \left[\begin{array}{rr} 2 & -5 \\ 1 & 0 \end{array}\right] = \left[\begin{array}{rr} 4 & -10 \\ 2 & 0 \end{array}\right] $[/tex]
The result is not among the matrices provided, so this operation doesn’t match any of our given results.
### Summary:
- [tex]$2A - B \longrightarrow \left[\begin{array}{rr}6 & -5 \\ 1 & 0\end{array}\right]$[/tex]
- [tex]$2A + 3A \longrightarrow \left[\begin{array}{cc}10 & -25 \\ 5 & 0\end{array}\right]$[/tex]
- [tex]$2(A - B) \longrightarrow \left[\begin{array}{ll}8 & 0 \\ 0 & 0\end{array}\right]$[/tex]
The matrices provided are:
[tex]$ A = \left[\begin{array}{rr} 2 & -5 \\ 1 & 0 \end{array}\right], \quad B = \left[\begin{array}{rr} -2 & -5 \\ 1 & 0 \end{array}\right] $[/tex]
The required matrix operations are:
- [tex]$2(A + B)$[/tex]
- [tex]$2A + 3A$[/tex]
- [tex]$2(A - B)$[/tex]
- [tex]$2A - B$[/tex]
- [tex]$2A$[/tex]
And the resulting matrices are:
- [tex]$\left[\begin{array}{rr}6 & -5 \\ 1 & 0\end{array}\right]$[/tex]
- [tex]$\left[\begin{array}{cc}10 & -25 \\ 5 & 0\end{array}\right]$[/tex]
- [tex]$\left[\begin{array}{ll}8 & 0 \\ 0 & 0\end{array}\right]$[/tex]
Let's match each matrix operation to its result.
### Matching Matrix Operations with Results:
1. [tex]$2(A + B)$[/tex]:
Calculate [tex]$A + B$[/tex]:
[tex]$ A + B = \left[\begin{array}{rr} 2 & -5 \\ 1 & 0 \end{array}\right] + \left[\begin{array}{rr} -2 & -5 \\ 1 & 0 \end{array}\right] = \left[\begin{array}{rr} 0 & -10 \\ 2 & 0 \end{array}\right] $[/tex]
Now multiply by 2:
[tex]$ 2(A + B) = 2 \times \left[\begin{array}{rr} 0 & -10 \\ 2 & 0 \end{array}\right] = \left[\begin{array}{rr} 0 & -20 \\ 4 & 0 \end{array}\right] $[/tex]
The result is not among the matrices provided, so this operation doesn’t match any of our given results.
2. [tex]$2A + 3A$[/tex]:
Calculate:
[tex]$ 2A + 3A = 5A = 5 \times \left[\begin{array}{rr} 2 & -5 \\ 1 & 0 \end{array}\right] = \left[\begin{array}{rr} 10 & -25 \\ 5 & 0 \end{array}\right] $[/tex]
Matching this matrix:
[tex]$ \left[\begin{array}{cc}10 & -25 \\ 5 & 0\end{array}\right] \longrightarrow 2A + 3A $[/tex]
3. [tex]$2(A - B)$[/tex]:
Calculate [tex]$A - B$[/tex]:
[tex]$ A - B = \left[\begin{array}{rr} 2 & -5 \\ 1 & 0 \end{array}\right] - \left[\begin{array}{rr} -2 & -5 \\ 1 & 0 \end{array}\right] = \left[\begin{array}{rr} 4 & 0 \\ 0 & 0 \end{array}\right] $[/tex]
Now multiply by 2:
[tex]$ 2(A - B) = 2 \times \left[\begin{array}{rr} 4 & 0 \\ 0 & 0 \end{array}\right] = \left[\begin{array}{rr} 8 & 0 \\ 0 & 0 \end{array}\right] $[/tex]
Matching this matrix:
[tex]$ \left[\begin{array}{ll}8 & 0 \\ 0 & 0\end{array}\right] \longrightarrow 2(A - B) $[/tex]
4. [tex]$2A - B$[/tex]:
Calculate:
[tex]$ 2A - B = 2 \times \left[\begin{array}{rr} 2 & -5 \\ 1 & 0 \end{array}\right] - \left[\begin{array}{rr} -2 & -5 \\ 1 & 0 \end{array}\right] = \left[\begin{array}{rr} 4 & -10 \\ 2 & 0 \end{array}\right] - \left[\begin{array}{rr} -2 & -5 \\ 1 & 0 \end{array}\right] = \left[\begin{array}{rr} 6 & -5 \\ 1 & 0 \end{array}\right] $[/tex]
Matching this matrix:
[tex]$ \left[\begin{array}{rr}6 & -5 \\ 1 & 0\end{array}\right] \longrightarrow 2A - B $[/tex]
5. [tex]$2A$[/tex]:
Calculate:
[tex]$ 2A = 2 \times \left[\begin{array}{rr} 2 & -5 \\ 1 & 0 \end{array}\right] = \left[\begin{array}{rr} 4 & -10 \\ 2 & 0 \end{array}\right] $[/tex]
The result is not among the matrices provided, so this operation doesn’t match any of our given results.
### Summary:
- [tex]$2A - B \longrightarrow \left[\begin{array}{rr}6 & -5 \\ 1 & 0\end{array}\right]$[/tex]
- [tex]$2A + 3A \longrightarrow \left[\begin{array}{cc}10 & -25 \\ 5 & 0\end{array}\right]$[/tex]
- [tex]$2(A - B) \longrightarrow \left[\begin{array}{ll}8 & 0 \\ 0 & 0\end{array}\right]$[/tex]
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