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Sagot :
Let's solve each of the given parts step-by-step using matrix operations.
Given the matrices:
[tex]\[ A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \][/tex]
[tex]\[ B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} \][/tex]
[tex]\[ C = \begin{bmatrix} 9 & 10 \\ 11 & 12 \end{bmatrix} \][/tex]
### a) [tex]\( A + B \)[/tex]
We perform element-wise addition of matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ A + B = \begin{bmatrix} 1+5 & 2+6 \\ 3+7 & 4+8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix} \][/tex]
### b) [tex]\( B - C \)[/tex]
We perform element-wise subtraction of matrix [tex]\( C \)[/tex] from [tex]\( B \)[/tex]:
[tex]\[ B - C = \begin{bmatrix} 5-9 & 6-10 \\ 7-11 & 8-12 \end{bmatrix} = \begin{bmatrix} -4 & -4 \\ -4 & -4 \end{bmatrix} \][/tex]
### c) [tex]\( C - B^t \)[/tex]
First, we find the transpose of [tex]\( B \)[/tex] ([tex]\( B^t \)[/tex]) and then subtract it from [tex]\( C \)[/tex]:
[tex]\[ B^t = \begin{bmatrix} 5 & 7 \\ 6 & 8 \end{bmatrix} \][/tex]
[tex]\[ C - B^t = \begin{bmatrix} 9-5 & 10-7 \\ 11-6 & 12-8 \end{bmatrix} = \begin{bmatrix} 4 & 3 \\ 5 & 4 \end{bmatrix} \][/tex]
### d) [tex]\( A + C^t \)[/tex]
First, we find the transpose of [tex]\( C \)[/tex] ([tex]\( C^t \)[/tex]) and then add it to [tex]\( A \)[/tex]:
[tex]\[ C^t = \begin{bmatrix} 9 & 11 \\ 10 & 12 \end{bmatrix} \][/tex]
[tex]\[ A + C^t = \begin{bmatrix} 1+9 & 2+10 \\ 3+11 & 4+12 \end{bmatrix} = \begin{bmatrix} 10 & 12 \\ 13 & 16 \end{bmatrix} \][/tex]
### e) [tex]\( A + B + C \)[/tex]
We perform element-wise addition of matrices [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex]:
[tex]\[ A + B + C = \begin{bmatrix} 1+5+9 & 2+6+10 \\ 3+7+11 & 4+8+12 \end{bmatrix} = \begin{bmatrix} 15 & 18 \\ 21 & 24 \end{bmatrix} \][/tex]
### f) [tex]\( A - B + C \)[/tex]
We perform element-wise subtraction of [tex]\( B \)[/tex] from [tex]\( A \)[/tex], followed by addition of [tex]\( C \)[/tex]:
[tex]\[ A - B + C = \begin{bmatrix} 1-5+9 & 2-6+10 \\ 3-7+11 & 4-8+12 \end{bmatrix} = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} \][/tex]
### g) [tex]\( A^t + C + B^t \)[/tex]
First, find the transposes of [tex]\( A \)[/tex] ([tex]\( A^t \)[/tex]) and [tex]\( B \)[/tex] ([tex]\( B^t \)[/tex]), then perform element-wise addition with [tex]\( C \)[/tex]:
[tex]\[ A^t = \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix} \][/tex]
[tex]\[ B^t = \begin{bmatrix} 5 & 7 \\ 6 & 8 \end{bmatrix} \][/tex]
[tex]\[ A^t + C + B^t = \begin{bmatrix} 1+9+5 & 3+11+7 \\ 2+10+6 & 4+12+8 \end{bmatrix} = \begin{bmatrix} 15 & 21 \\ 18 & 24 \end{bmatrix} \][/tex]
### h) [tex]\( -A + B - C^t \)[/tex]
First, find the transpose of [tex]\( C \)[/tex] ([tex]\( C^t \)[/tex]), negate the elements of [tex]\( A \)[/tex], and then perform element-wise subtraction of [tex]\( C^t \)[/tex] from [tex]\( B \)[/tex] and addition to [tex]\(-A\)[/tex]:
[tex]\[ C^t = \begin{bmatrix} 9 & 11 \\ 10 & 12 \end{bmatrix} \][/tex]
[tex]\[ -A = \begin{bmatrix} -1 & -2 \\ -3 & -4 \end{bmatrix} \][/tex]
[tex]\[ -A + B - C^t = \begin{bmatrix} -1+5-9 & -2+6-11 \\ -3+7-10 & -4+8-12 \end{bmatrix} = \begin{bmatrix} -5 & -7 \\ -6 & -8 \end{bmatrix} \][/tex]
In conclusion, the calculated results for each part are:
a) [tex]\( \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix} \)[/tex]
b) [tex]\( \begin{bmatrix} -4 & -4 \\ -4 & -4 \end{bmatrix} \)[/tex]
c) [tex]\( \begin{bmatrix} 4 & 3 \\ 5 & 4 \end{bmatrix} \)[/tex]
d) [tex]\( \begin{bmatrix} 10 & 12 \\ 13 & 16 \end{bmatrix} \)[/tex]
e) [tex]\( \begin{bmatrix} 15 & 18 \\ 21 & 24 \end{bmatrix} \)[/tex]
f) [tex]\( \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} \)[/tex]
g) [tex]\( \begin{bmatrix} 15 & 21 \\ 18 & 24 \end{bmatrix} \)[/tex]
h) [tex]\( \begin{bmatrix} -5 & -7 \\ -6 & -8 \end{bmatrix} \)[/tex]
Given the matrices:
[tex]\[ A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \][/tex]
[tex]\[ B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} \][/tex]
[tex]\[ C = \begin{bmatrix} 9 & 10 \\ 11 & 12 \end{bmatrix} \][/tex]
### a) [tex]\( A + B \)[/tex]
We perform element-wise addition of matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ A + B = \begin{bmatrix} 1+5 & 2+6 \\ 3+7 & 4+8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix} \][/tex]
### b) [tex]\( B - C \)[/tex]
We perform element-wise subtraction of matrix [tex]\( C \)[/tex] from [tex]\( B \)[/tex]:
[tex]\[ B - C = \begin{bmatrix} 5-9 & 6-10 \\ 7-11 & 8-12 \end{bmatrix} = \begin{bmatrix} -4 & -4 \\ -4 & -4 \end{bmatrix} \][/tex]
### c) [tex]\( C - B^t \)[/tex]
First, we find the transpose of [tex]\( B \)[/tex] ([tex]\( B^t \)[/tex]) and then subtract it from [tex]\( C \)[/tex]:
[tex]\[ B^t = \begin{bmatrix} 5 & 7 \\ 6 & 8 \end{bmatrix} \][/tex]
[tex]\[ C - B^t = \begin{bmatrix} 9-5 & 10-7 \\ 11-6 & 12-8 \end{bmatrix} = \begin{bmatrix} 4 & 3 \\ 5 & 4 \end{bmatrix} \][/tex]
### d) [tex]\( A + C^t \)[/tex]
First, we find the transpose of [tex]\( C \)[/tex] ([tex]\( C^t \)[/tex]) and then add it to [tex]\( A \)[/tex]:
[tex]\[ C^t = \begin{bmatrix} 9 & 11 \\ 10 & 12 \end{bmatrix} \][/tex]
[tex]\[ A + C^t = \begin{bmatrix} 1+9 & 2+10 \\ 3+11 & 4+12 \end{bmatrix} = \begin{bmatrix} 10 & 12 \\ 13 & 16 \end{bmatrix} \][/tex]
### e) [tex]\( A + B + C \)[/tex]
We perform element-wise addition of matrices [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex]:
[tex]\[ A + B + C = \begin{bmatrix} 1+5+9 & 2+6+10 \\ 3+7+11 & 4+8+12 \end{bmatrix} = \begin{bmatrix} 15 & 18 \\ 21 & 24 \end{bmatrix} \][/tex]
### f) [tex]\( A - B + C \)[/tex]
We perform element-wise subtraction of [tex]\( B \)[/tex] from [tex]\( A \)[/tex], followed by addition of [tex]\( C \)[/tex]:
[tex]\[ A - B + C = \begin{bmatrix} 1-5+9 & 2-6+10 \\ 3-7+11 & 4-8+12 \end{bmatrix} = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} \][/tex]
### g) [tex]\( A^t + C + B^t \)[/tex]
First, find the transposes of [tex]\( A \)[/tex] ([tex]\( A^t \)[/tex]) and [tex]\( B \)[/tex] ([tex]\( B^t \)[/tex]), then perform element-wise addition with [tex]\( C \)[/tex]:
[tex]\[ A^t = \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix} \][/tex]
[tex]\[ B^t = \begin{bmatrix} 5 & 7 \\ 6 & 8 \end{bmatrix} \][/tex]
[tex]\[ A^t + C + B^t = \begin{bmatrix} 1+9+5 & 3+11+7 \\ 2+10+6 & 4+12+8 \end{bmatrix} = \begin{bmatrix} 15 & 21 \\ 18 & 24 \end{bmatrix} \][/tex]
### h) [tex]\( -A + B - C^t \)[/tex]
First, find the transpose of [tex]\( C \)[/tex] ([tex]\( C^t \)[/tex]), negate the elements of [tex]\( A \)[/tex], and then perform element-wise subtraction of [tex]\( C^t \)[/tex] from [tex]\( B \)[/tex] and addition to [tex]\(-A\)[/tex]:
[tex]\[ C^t = \begin{bmatrix} 9 & 11 \\ 10 & 12 \end{bmatrix} \][/tex]
[tex]\[ -A = \begin{bmatrix} -1 & -2 \\ -3 & -4 \end{bmatrix} \][/tex]
[tex]\[ -A + B - C^t = \begin{bmatrix} -1+5-9 & -2+6-11 \\ -3+7-10 & -4+8-12 \end{bmatrix} = \begin{bmatrix} -5 & -7 \\ -6 & -8 \end{bmatrix} \][/tex]
In conclusion, the calculated results for each part are:
a) [tex]\( \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix} \)[/tex]
b) [tex]\( \begin{bmatrix} -4 & -4 \\ -4 & -4 \end{bmatrix} \)[/tex]
c) [tex]\( \begin{bmatrix} 4 & 3 \\ 5 & 4 \end{bmatrix} \)[/tex]
d) [tex]\( \begin{bmatrix} 10 & 12 \\ 13 & 16 \end{bmatrix} \)[/tex]
e) [tex]\( \begin{bmatrix} 15 & 18 \\ 21 & 24 \end{bmatrix} \)[/tex]
f) [tex]\( \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} \)[/tex]
g) [tex]\( \begin{bmatrix} 15 & 21 \\ 18 & 24 \end{bmatrix} \)[/tex]
h) [tex]\( \begin{bmatrix} -5 & -7 \\ -6 & -8 \end{bmatrix} \)[/tex]
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