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Sagot :
To find the given matrix operations, we will systematically go through each required calculation.
1. Matrix Addition ([tex]$A + B$[/tex])
The matrices given are:
[tex]\( A = \left[\begin{array}{cc} 2 & 1 \\ 1 & 2 \end{array}\right] \)[/tex]
[tex]\( B = \left[\begin{array}{rr} -4 & -4 \\ 1 & 4 \end{array}\right] \)[/tex]
To add matrices [tex]\(A\)[/tex] and [tex]\(B\)[/tex], we add corresponding elements:
[tex]\[ A + B = \left[\begin{array}{cc} 2 & 1 \\ 1 & 2 \end{array}\right] + \left[\begin{array}{rr} -4 & -4 \\ 1 & 4 \end{array}\right] = \left[\begin{array}{cc} 2 + (-4) & 1 + (-4) \\ 1 + 1 & 2 + 4 \end{array}\right] \][/tex]
Simplifying, we get:
[tex]\[ A + B = \left[\begin{array}{cc} -2 & -3 \\ 2 & 6 \end{array}\right] \][/tex]
2. Matrix Subtraction ([tex]$A - B$[/tex])
To subtract matrix [tex]\(B\)[/tex] from [tex]\(A\)[/tex], we subtract corresponding elements:
[tex]\[ A - B = \left[\begin{array}{cc} 2 & 1 \\ 1 & 2 \end{array}\right] - \left[\begin{array}{rr} -4 & -4 \\ 1 & 4 \end{array}\right] = \left[\begin{array}{cc} 2 - (-4) & 1 - (-4) \\ 1 - 1 & 2 - 4 \end{array}\right] \][/tex]
Simplifying, we get:
[tex]\[ A - B = \left[\begin{array}{cc} 6 & 5 \\ 0 & -2 \end{array}\right] \][/tex]
3. Scalar Multiplication ([tex]$4A$[/tex])
To calculate [tex]\(4A\)[/tex], we multiply all elements of matrix [tex]\(A\)[/tex] by 4:
[tex]\[ 4A = 4 \times \left[\begin{array}{cc} 2 & 1 \\ 1 & 2 \end{array}\right] = \left[\begin{array}{cc} 4 \times 2 & 4 \times 1 \\ 4 \times 1 & 4 \times 2 \end{array}\right] \][/tex]
Simplifying, we get:
[tex]\[ 4A = \left[\begin{array}{cc} 8 & 4 \\ 4 & 8 \end{array}\right] \][/tex]
4. Linear Combination ([tex]$4A - 5B$[/tex])
First, we compute [tex]\(5B\)[/tex]:
[tex]\[ 5B = 5 \times \left[\begin{array}{rr} -4 & -4 \\ 1 & 4 \end{array}\right] = \left[\begin{array}{rr} 5 \times -4 & 5 \times -4 \\ 5 \times 1 & 5 \times 4 \end{array}\right] = \left[\begin{array}{rr} -20 & -20 \\ 5 & 20 \end{array}\right] \][/tex]
Next, we calculate [tex]\(4A - 5B\)[/tex]:
[tex]\[ 4A - 5B = \left[\begin{array}{cc} 8 & 4 \\ 4 & 8 \end{array}\right] - \left[\begin{array}{rr} -20 & -20 \\ 5 & 20 \end{array}\right] = \left[\begin{array}{cc} 8 - (-20) & 4 - (-20) \\ 4 - 5 & 8 - 20 \end{array}\right] \][/tex]
Simplifying, we get:
[tex]\[ 4A - 5B = \left[\begin{array}{cc} 28 & 24 \\ -1 & -12 \end{array}\right] \][/tex]
Thus, the results are:
(a) [tex]\(A + B = \left[\begin{array}{cc} -2 & -3 \\ 2 & 6 \end{array}\right]\)[/tex]
(b) [tex]\(A - B = \left[\begin{array}{cc} 6 & 5 \\ 0 & -2 \end{array}\right]\)[/tex]
(c) [tex]\(4A = \left[\begin{array}{cc} 8 & 4 \\ 4 & 8 \end{array}\right]\)[/tex]
(d) [tex]\(4A - 5B = \left[\begin{array}{cc} 28 & 24 \\ -1 & -12 \end{array}\right]\)[/tex]
1. Matrix Addition ([tex]$A + B$[/tex])
The matrices given are:
[tex]\( A = \left[\begin{array}{cc} 2 & 1 \\ 1 & 2 \end{array}\right] \)[/tex]
[tex]\( B = \left[\begin{array}{rr} -4 & -4 \\ 1 & 4 \end{array}\right] \)[/tex]
To add matrices [tex]\(A\)[/tex] and [tex]\(B\)[/tex], we add corresponding elements:
[tex]\[ A + B = \left[\begin{array}{cc} 2 & 1 \\ 1 & 2 \end{array}\right] + \left[\begin{array}{rr} -4 & -4 \\ 1 & 4 \end{array}\right] = \left[\begin{array}{cc} 2 + (-4) & 1 + (-4) \\ 1 + 1 & 2 + 4 \end{array}\right] \][/tex]
Simplifying, we get:
[tex]\[ A + B = \left[\begin{array}{cc} -2 & -3 \\ 2 & 6 \end{array}\right] \][/tex]
2. Matrix Subtraction ([tex]$A - B$[/tex])
To subtract matrix [tex]\(B\)[/tex] from [tex]\(A\)[/tex], we subtract corresponding elements:
[tex]\[ A - B = \left[\begin{array}{cc} 2 & 1 \\ 1 & 2 \end{array}\right] - \left[\begin{array}{rr} -4 & -4 \\ 1 & 4 \end{array}\right] = \left[\begin{array}{cc} 2 - (-4) & 1 - (-4) \\ 1 - 1 & 2 - 4 \end{array}\right] \][/tex]
Simplifying, we get:
[tex]\[ A - B = \left[\begin{array}{cc} 6 & 5 \\ 0 & -2 \end{array}\right] \][/tex]
3. Scalar Multiplication ([tex]$4A$[/tex])
To calculate [tex]\(4A\)[/tex], we multiply all elements of matrix [tex]\(A\)[/tex] by 4:
[tex]\[ 4A = 4 \times \left[\begin{array}{cc} 2 & 1 \\ 1 & 2 \end{array}\right] = \left[\begin{array}{cc} 4 \times 2 & 4 \times 1 \\ 4 \times 1 & 4 \times 2 \end{array}\right] \][/tex]
Simplifying, we get:
[tex]\[ 4A = \left[\begin{array}{cc} 8 & 4 \\ 4 & 8 \end{array}\right] \][/tex]
4. Linear Combination ([tex]$4A - 5B$[/tex])
First, we compute [tex]\(5B\)[/tex]:
[tex]\[ 5B = 5 \times \left[\begin{array}{rr} -4 & -4 \\ 1 & 4 \end{array}\right] = \left[\begin{array}{rr} 5 \times -4 & 5 \times -4 \\ 5 \times 1 & 5 \times 4 \end{array}\right] = \left[\begin{array}{rr} -20 & -20 \\ 5 & 20 \end{array}\right] \][/tex]
Next, we calculate [tex]\(4A - 5B\)[/tex]:
[tex]\[ 4A - 5B = \left[\begin{array}{cc} 8 & 4 \\ 4 & 8 \end{array}\right] - \left[\begin{array}{rr} -20 & -20 \\ 5 & 20 \end{array}\right] = \left[\begin{array}{cc} 8 - (-20) & 4 - (-20) \\ 4 - 5 & 8 - 20 \end{array}\right] \][/tex]
Simplifying, we get:
[tex]\[ 4A - 5B = \left[\begin{array}{cc} 28 & 24 \\ -1 & -12 \end{array}\right] \][/tex]
Thus, the results are:
(a) [tex]\(A + B = \left[\begin{array}{cc} -2 & -3 \\ 2 & 6 \end{array}\right]\)[/tex]
(b) [tex]\(A - B = \left[\begin{array}{cc} 6 & 5 \\ 0 & -2 \end{array}\right]\)[/tex]
(c) [tex]\(4A = \left[\begin{array}{cc} 8 & 4 \\ 4 & 8 \end{array}\right]\)[/tex]
(d) [tex]\(4A - 5B = \left[\begin{array}{cc} 28 & 24 \\ -1 & -12 \end{array}\right]\)[/tex]
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