Get personalized answers to your specific questions with IDNLearn.com. Join our platform to receive prompt and accurate responses from experienced professionals in various fields.
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]
Thank you for using this platform to share and learn. Don't hesitate to keep asking and answering. We value every contribution you make. Thank you for trusting IDNLearn.com. We’re dedicated to providing accurate answers, so visit us again for more solutions.
