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Sagot :
Sure, let's go through each part of the question step by step.
### Part (a): Finding the matrix [tex]\(2A + B\)[/tex]
Given matrices:
[tex]\[ A = \begin{pmatrix} 3 & 4 \\ 5 & -2 \end{pmatrix} \][/tex]
[tex]\[ B = \begin{pmatrix} 1 & -4 \\ 0 & 1 \end{pmatrix} \][/tex]
First, calculate [tex]\(2A\)[/tex]:
[tex]\[ 2A = 2 \cdot \begin{pmatrix} 3 & 4 \\ 5 & -2 \end{pmatrix} = \begin{pmatrix} 6 & 8 \\ 10 & -4 \end{pmatrix} \][/tex]
Next, add [tex]\(B\)[/tex] to [tex]\(2A\)[/tex]:
[tex]\[ 2A + B = \begin{pmatrix} 6 & 8 \\ 10 & -4 \end{pmatrix} + \begin{pmatrix} 1 & -4 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 7 & 4 \\ 10 & -3 \end{pmatrix} \][/tex]
So, the matrix [tex]\(2A + B\)[/tex] is:
[tex]\[ \begin{pmatrix} 7 & 4 \\ 10 & -3 \end{pmatrix} \][/tex]
### Part (b): Solving for [tex]\(A\)[/tex]
Given the matrix equation:
[tex]\[ \begin{pmatrix} 2 & -1 \\ 2 & 0 \end{pmatrix} + 2A = \begin{pmatrix} -3 & 5 \\ 4 & 3 \end{pmatrix} \][/tex]
First, isolate [tex]\(2A\)[/tex] by subtracting [tex]\(\begin{pmatrix} 2 & -1 \\ 2 & 0 \end{pmatrix}\)[/tex] from both sides:
[tex]\[ 2A = \begin{pmatrix} -3 & 5 \\ 4 & 3 \end{pmatrix} - \begin{pmatrix} 2 & -1 \\ 2 & 0 \end{pmatrix} = \begin{pmatrix} -5 & 6 \\ 2 & 3 \end{pmatrix} \][/tex]
Next, divide by 2 to solve for [tex]\(A\)[/tex]:
[tex]\[ A = \frac{1}{2} \begin{pmatrix} -5 & 6 \\ 2 & 3 \end{pmatrix} = \begin{pmatrix} -2.5 & 3 \\ 1 & 1.5 \end{pmatrix} \][/tex]
So, the matrix [tex]\(A\)[/tex] is:
[tex]\[ \begin{pmatrix} -2.5 & 3 \\ 1 & 1.5 \end{pmatrix} \][/tex]
### Part (c): Finding the value of [tex]\(y\)[/tex]
Given the matrix equation:
[tex]\[ 2 \begin{pmatrix} 3 & 4 \\ 5 & -2 \end{pmatrix} + \begin{pmatrix} 1 & y \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 7 & 0 \\ 10 & -3 \end{pmatrix} \][/tex]
First, calculate [tex]\(2 \begin{pmatrix} 3 & 4 \\ 5 & -2 \end{pmatrix}\)[/tex]:
[tex]\[ 2 \begin{pmatrix} 3 & 4 \\ 5 & -2 \end{pmatrix} = \begin{pmatrix} 6 & 8 \\ 10 & -4 \end{pmatrix} \][/tex]
Next, set up the matrix equation:
[tex]\[ \begin{pmatrix} 6 & 8 \\ 10 & -4 \end{pmatrix} + \begin{pmatrix} 1 & y \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 7 & 0 \\ 10 & -3 \end{pmatrix} \][/tex]
Equate the left side to the right side:
[tex]\[ \begin{pmatrix} 6 + 1 & 8 + y \\ 10 + 0 & -4 + 1 \end{pmatrix} = \begin{pmatrix} 7 & 0 \\ 10 & -3 \end{pmatrix} \][/tex]
From the matrix equality, we see that:
[tex]\[ 8 + y = 0 \][/tex]
Solving for [tex]\(y\)[/tex]:
[tex]\[ y = -8 \][/tex]
Thus, the value of [tex]\(y\)[/tex] is:
[tex]\[ y = -8 \][/tex]
In summary, the answers are:
(a) [tex]\[ \begin{pmatrix} 7 & 4 \\ 10 & -3 \end{pmatrix} \][/tex]
(b) [tex]\[ \begin{pmatrix} -2.5 & 3 \\ 1 & 1.5 \end{pmatrix} \][/tex]
(c) [tex]\( y = -8 \)[/tex]
### Part (a): Finding the matrix [tex]\(2A + B\)[/tex]
Given matrices:
[tex]\[ A = \begin{pmatrix} 3 & 4 \\ 5 & -2 \end{pmatrix} \][/tex]
[tex]\[ B = \begin{pmatrix} 1 & -4 \\ 0 & 1 \end{pmatrix} \][/tex]
First, calculate [tex]\(2A\)[/tex]:
[tex]\[ 2A = 2 \cdot \begin{pmatrix} 3 & 4 \\ 5 & -2 \end{pmatrix} = \begin{pmatrix} 6 & 8 \\ 10 & -4 \end{pmatrix} \][/tex]
Next, add [tex]\(B\)[/tex] to [tex]\(2A\)[/tex]:
[tex]\[ 2A + B = \begin{pmatrix} 6 & 8 \\ 10 & -4 \end{pmatrix} + \begin{pmatrix} 1 & -4 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 7 & 4 \\ 10 & -3 \end{pmatrix} \][/tex]
So, the matrix [tex]\(2A + B\)[/tex] is:
[tex]\[ \begin{pmatrix} 7 & 4 \\ 10 & -3 \end{pmatrix} \][/tex]
### Part (b): Solving for [tex]\(A\)[/tex]
Given the matrix equation:
[tex]\[ \begin{pmatrix} 2 & -1 \\ 2 & 0 \end{pmatrix} + 2A = \begin{pmatrix} -3 & 5 \\ 4 & 3 \end{pmatrix} \][/tex]
First, isolate [tex]\(2A\)[/tex] by subtracting [tex]\(\begin{pmatrix} 2 & -1 \\ 2 & 0 \end{pmatrix}\)[/tex] from both sides:
[tex]\[ 2A = \begin{pmatrix} -3 & 5 \\ 4 & 3 \end{pmatrix} - \begin{pmatrix} 2 & -1 \\ 2 & 0 \end{pmatrix} = \begin{pmatrix} -5 & 6 \\ 2 & 3 \end{pmatrix} \][/tex]
Next, divide by 2 to solve for [tex]\(A\)[/tex]:
[tex]\[ A = \frac{1}{2} \begin{pmatrix} -5 & 6 \\ 2 & 3 \end{pmatrix} = \begin{pmatrix} -2.5 & 3 \\ 1 & 1.5 \end{pmatrix} \][/tex]
So, the matrix [tex]\(A\)[/tex] is:
[tex]\[ \begin{pmatrix} -2.5 & 3 \\ 1 & 1.5 \end{pmatrix} \][/tex]
### Part (c): Finding the value of [tex]\(y\)[/tex]
Given the matrix equation:
[tex]\[ 2 \begin{pmatrix} 3 & 4 \\ 5 & -2 \end{pmatrix} + \begin{pmatrix} 1 & y \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 7 & 0 \\ 10 & -3 \end{pmatrix} \][/tex]
First, calculate [tex]\(2 \begin{pmatrix} 3 & 4 \\ 5 & -2 \end{pmatrix}\)[/tex]:
[tex]\[ 2 \begin{pmatrix} 3 & 4 \\ 5 & -2 \end{pmatrix} = \begin{pmatrix} 6 & 8 \\ 10 & -4 \end{pmatrix} \][/tex]
Next, set up the matrix equation:
[tex]\[ \begin{pmatrix} 6 & 8 \\ 10 & -4 \end{pmatrix} + \begin{pmatrix} 1 & y \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 7 & 0 \\ 10 & -3 \end{pmatrix} \][/tex]
Equate the left side to the right side:
[tex]\[ \begin{pmatrix} 6 + 1 & 8 + y \\ 10 + 0 & -4 + 1 \end{pmatrix} = \begin{pmatrix} 7 & 0 \\ 10 & -3 \end{pmatrix} \][/tex]
From the matrix equality, we see that:
[tex]\[ 8 + y = 0 \][/tex]
Solving for [tex]\(y\)[/tex]:
[tex]\[ y = -8 \][/tex]
Thus, the value of [tex]\(y\)[/tex] is:
[tex]\[ y = -8 \][/tex]
In summary, the answers are:
(a) [tex]\[ \begin{pmatrix} 7 & 4 \\ 10 & -3 \end{pmatrix} \][/tex]
(b) [tex]\[ \begin{pmatrix} -2.5 & 3 \\ 1 & 1.5 \end{pmatrix} \][/tex]
(c) [tex]\( y = -8 \)[/tex]
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