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Certainly! Let's go through the questions one by one with detailed steps.
### Part 1: Finding [tex]\( A + B \)[/tex] and [tex]\( A - B \)[/tex]
Given matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ A = \begin{bmatrix} 2 & 5 \\ 4 & 5 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & 4 \\ 0 & 8 \end{bmatrix} \][/tex]
To find [tex]\( A + B \)[/tex]:
[tex]\[ A + B = \begin{bmatrix} 2 & 5 \\ 4 & 5 \end{bmatrix} + \begin{bmatrix} 2 & 4 \\ 0 & 8 \end{bmatrix} \][/tex]
We add corresponding elements of matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ A + B = \begin{bmatrix} 2 + 2 & 5 + 4 \\ 4 + 0 & 5 + 8 \end{bmatrix} = \begin{bmatrix} 4 & 9 \\ 4 & 13 \end{bmatrix} \][/tex]
So, [tex]\( A + B \)[/tex] is:
[tex]\[ \begin{bmatrix} 4 & 9 \\ 4 & 13 \end{bmatrix} \][/tex]
To find [tex]\( A - B \)[/tex]:
[tex]\[ A - B = \begin{bmatrix} 2 & 5 \\ 4 & 5 \end{bmatrix} - \begin{bmatrix} 2 & 4 \\ 0 & 8 \end{bmatrix} \][/tex]
We subtract corresponding elements of matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ A - B = \begin{bmatrix} 2 - 2 & 5 - 4 \\ 4 - 0 & 5 - 8 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 4 & -3 \end{bmatrix} \][/tex]
So, [tex]\( A - B \)[/tex] is:
[tex]\[ \begin{bmatrix} 0 & 1 \\ 4 & -3 \end{bmatrix} \][/tex]
### Part 2: Finding [tex]\( 5A + 5B \)[/tex]
Given matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ A = \begin{bmatrix} 3 & 1 & 2 \\ 0 & 8 & 7 \\ 9 & 11 & -4 \end{bmatrix}, \quad B = \begin{bmatrix} 7 & 1 & 9 \\ 3 & 6 & -1 \\ 4 & -6 & 12 \end{bmatrix} \][/tex]
To find [tex]\( 5A + 5B \)[/tex]:
Firstly, calculate [tex]\( 5A \)[/tex] by multiplying all elements of matrix [tex]\( A \)[/tex] by 5.
[tex]\[ 5A = 5 \times \begin{bmatrix} 3 & 1 & 2 \\ 0 & 8 & 7 \\ 9 & 11 & -4 \end{bmatrix} = \begin{bmatrix} 15 & 5 & 10 \\ 0 & 40 & 35 \\ 45 & 55 & -20 \end{bmatrix} \][/tex]
Next, calculate [tex]\( 5B \)[/tex] by multiplying all elements of matrix [tex]\( B \)[/tex] by 5.
[tex]\[ 5B = 5 \times \begin{bmatrix} 7 & 1 & 9 \\ 3 & 6 & -1 \\ 4 & -6 & 12 \end{bmatrix} = \begin{bmatrix} 35 & 5 & 45 \\ 15 & 30 & -5 \\ 20 & -30 & 60 \end{bmatrix} \][/tex]
Now, add the resulting matrices [tex]\( 5A \)[/tex] and [tex]\( 5B \)[/tex]:
[tex]\[ 5A + 5B = \begin{bmatrix} 15 & 5 & 10 \\ 0 & 40 & 35 \\ 45 & 55 & -20 \end{bmatrix} + \begin{bmatrix} 35 & 5 & 45 \\ 15 & 30 & -5 \\ 20 & -30 & 60 \end{bmatrix} \][/tex]
Add corresponding elements:
[tex]\[ 5A + 5B = \begin{bmatrix} 15 + 35 & 5 + 5 & 10 + 45 \\ 0 + 15 & 40 + 30 & 35 + -5 \\ 45 + 20 & 55 + -30 & -20 + 60 \end{bmatrix} = \begin{bmatrix} 50 & 10 & 55 \\ 15 & 70 & 30 \\ 65 & 25 & 40 \end{bmatrix} \][/tex]
So, [tex]\( 5A + 5B \)[/tex] is:
[tex]\[ \begin{bmatrix} 50 & 10 & 55 \\ 15 & 70 & 30 \\ 65 & 25 & 40 \end{bmatrix} \][/tex]
In summary:
1. [tex]\( A + B = \begin{bmatrix} 4 & 9 \\ 4 & 13 \end{bmatrix} \)[/tex],
[tex]\( A - B = \begin{bmatrix} 0 & 1 \\ 4 & -3 \end{bmatrix} \)[/tex].
2. [tex]\( 5A + 5B = \begin{bmatrix} 50 & 10 & 55 \\ 15 & 70 & 30 \\ 65 & 25 & 40 \end{bmatrix} \)[/tex].
### Part 1: Finding [tex]\( A + B \)[/tex] and [tex]\( A - B \)[/tex]
Given matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ A = \begin{bmatrix} 2 & 5 \\ 4 & 5 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & 4 \\ 0 & 8 \end{bmatrix} \][/tex]
To find [tex]\( A + B \)[/tex]:
[tex]\[ A + B = \begin{bmatrix} 2 & 5 \\ 4 & 5 \end{bmatrix} + \begin{bmatrix} 2 & 4 \\ 0 & 8 \end{bmatrix} \][/tex]
We add corresponding elements of matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ A + B = \begin{bmatrix} 2 + 2 & 5 + 4 \\ 4 + 0 & 5 + 8 \end{bmatrix} = \begin{bmatrix} 4 & 9 \\ 4 & 13 \end{bmatrix} \][/tex]
So, [tex]\( A + B \)[/tex] is:
[tex]\[ \begin{bmatrix} 4 & 9 \\ 4 & 13 \end{bmatrix} \][/tex]
To find [tex]\( A - B \)[/tex]:
[tex]\[ A - B = \begin{bmatrix} 2 & 5 \\ 4 & 5 \end{bmatrix} - \begin{bmatrix} 2 & 4 \\ 0 & 8 \end{bmatrix} \][/tex]
We subtract corresponding elements of matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ A - B = \begin{bmatrix} 2 - 2 & 5 - 4 \\ 4 - 0 & 5 - 8 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 4 & -3 \end{bmatrix} \][/tex]
So, [tex]\( A - B \)[/tex] is:
[tex]\[ \begin{bmatrix} 0 & 1 \\ 4 & -3 \end{bmatrix} \][/tex]
### Part 2: Finding [tex]\( 5A + 5B \)[/tex]
Given matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ A = \begin{bmatrix} 3 & 1 & 2 \\ 0 & 8 & 7 \\ 9 & 11 & -4 \end{bmatrix}, \quad B = \begin{bmatrix} 7 & 1 & 9 \\ 3 & 6 & -1 \\ 4 & -6 & 12 \end{bmatrix} \][/tex]
To find [tex]\( 5A + 5B \)[/tex]:
Firstly, calculate [tex]\( 5A \)[/tex] by multiplying all elements of matrix [tex]\( A \)[/tex] by 5.
[tex]\[ 5A = 5 \times \begin{bmatrix} 3 & 1 & 2 \\ 0 & 8 & 7 \\ 9 & 11 & -4 \end{bmatrix} = \begin{bmatrix} 15 & 5 & 10 \\ 0 & 40 & 35 \\ 45 & 55 & -20 \end{bmatrix} \][/tex]
Next, calculate [tex]\( 5B \)[/tex] by multiplying all elements of matrix [tex]\( B \)[/tex] by 5.
[tex]\[ 5B = 5 \times \begin{bmatrix} 7 & 1 & 9 \\ 3 & 6 & -1 \\ 4 & -6 & 12 \end{bmatrix} = \begin{bmatrix} 35 & 5 & 45 \\ 15 & 30 & -5 \\ 20 & -30 & 60 \end{bmatrix} \][/tex]
Now, add the resulting matrices [tex]\( 5A \)[/tex] and [tex]\( 5B \)[/tex]:
[tex]\[ 5A + 5B = \begin{bmatrix} 15 & 5 & 10 \\ 0 & 40 & 35 \\ 45 & 55 & -20 \end{bmatrix} + \begin{bmatrix} 35 & 5 & 45 \\ 15 & 30 & -5 \\ 20 & -30 & 60 \end{bmatrix} \][/tex]
Add corresponding elements:
[tex]\[ 5A + 5B = \begin{bmatrix} 15 + 35 & 5 + 5 & 10 + 45 \\ 0 + 15 & 40 + 30 & 35 + -5 \\ 45 + 20 & 55 + -30 & -20 + 60 \end{bmatrix} = \begin{bmatrix} 50 & 10 & 55 \\ 15 & 70 & 30 \\ 65 & 25 & 40 \end{bmatrix} \][/tex]
So, [tex]\( 5A + 5B \)[/tex] is:
[tex]\[ \begin{bmatrix} 50 & 10 & 55 \\ 15 & 70 & 30 \\ 65 & 25 & 40 \end{bmatrix} \][/tex]
In summary:
1. [tex]\( A + B = \begin{bmatrix} 4 & 9 \\ 4 & 13 \end{bmatrix} \)[/tex],
[tex]\( A - B = \begin{bmatrix} 0 & 1 \\ 4 & -3 \end{bmatrix} \)[/tex].
2. [tex]\( 5A + 5B = \begin{bmatrix} 50 & 10 & 55 \\ 15 & 70 & 30 \\ 65 & 25 & 40 \end{bmatrix} \)[/tex].
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