IDNLearn.com makes it easy to get reliable answers from knowledgeable individuals. Our Q&A platform offers reliable and thorough answers to ensure you have the information you need to succeed in any situation.
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]
We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. IDNLearn.com provides the best answers to your questions. Thank you for visiting, and come back soon for more helpful information.