Certainly! Let's solve the given problem step-by-step.
Given matrices:
[tex]\[ A = \begin{bmatrix} 3 & 4 \\ -5 & -1 \end{bmatrix} \][/tex]
[tex]\[ 3A + B = \begin{bmatrix} 2 & 8 \\ 3 & -4 \end{bmatrix} \][/tex]
### i) Finding Matrix [tex]\(B\)[/tex]
First, isolate [tex]\(B\)[/tex] from the equation [tex]\(3A + B = \text{some matrix}\)[/tex]:
[tex]\[
B = \begin{bmatrix} 2 & 8 \\ 3 & -4 \end{bmatrix} - 3A
\][/tex]
Next, calculate [tex]\(3A\)[/tex]:
[tex]\[
3A = 3 \times \begin{bmatrix} 3 & 4 \\ -5 & -1 \end{bmatrix} = \begin{bmatrix} 3 \cdot 3 & 3 \cdot 4 \\ 3 \cdot (-5) & 3 \cdot (-1) \end{bmatrix} = \begin{bmatrix} 9 & 12 \\ -15 & -3 \end{bmatrix}
\][/tex]
Subtract [tex]\(3A\)[/tex] from the given matrix:
[tex]\[
B = \begin{bmatrix} 2 & 8 \\ 3 & -4 \end{bmatrix} - \begin{bmatrix} 9 & 12 \\ -15 & -3 \end{bmatrix} = \begin{bmatrix} 2 - 9 & 8 - 12 \\ 3 - (-15) & -4 - (-3) \end{bmatrix} = \begin{bmatrix} -7 & -4 \\ 18 & -1 \end{bmatrix}
\][/tex]
Thus, matrix [tex]\(B\)[/tex] is:
[tex]\[
B = \begin{bmatrix} -7 & -4 \\ 18 & -1 \end{bmatrix}
\][/tex]
### ii) Finding [tex]\(AB\)[/tex]
Now, we need to find the product [tex]\(AB\)[/tex]:
[tex]\[
A = \begin{bmatrix} 3 & 4 \\ -5 & -1 \end{bmatrix}
\][/tex]
[tex]\[
B = \begin{bmatrix} -7 & -4 \\ 18 & -1 \end{bmatrix}
\][/tex]
To multiply two matrices, we follow the rule where each element of the resulting matrix is the dot product of corresponding rows and columns.
So,
[tex]\[
AB = \begin{bmatrix} 3 & 4 \\ -5 & -1 \end{bmatrix} \times \begin{bmatrix} -7 & -4 \\ 18 & -1 \end{bmatrix}
\][/tex]
Calculate each element of the resulting matrix:
1. First row, first column:
[tex]\[
(3 \times -7) + (4 \times 18) = -21 + 72 = 51
\][/tex]
2. First row, second column:
[tex]\[
(3 \times -4) + (4 \times -1) = -12 - 4 = -16
\][/tex]
3. Second row, first column:
[tex]\[
(-5 \times -7) + (-1 \times 18) = 35 - 18 = 17
\][/tex]
4. Second row, second column:
[tex]\[
(-5 \times -4) + (-1 \times -1) = 20 + 1 = 21
\][/tex]
Thus, the resulting matrix [tex]\(AB\)[/tex] is:
[tex]\[
AB = \begin{bmatrix} 51 & -16 \\ 17 & 21 \end{bmatrix}
\][/tex]
### Summary
1. Matrix [tex]\(B\)[/tex] is:
[tex]\[
B = \begin{bmatrix} -7 & -4 \\ 18 & -1 \end{bmatrix}
\][/tex]
2. The product [tex]\(AB\)[/tex] is:
[tex]\[
AB = \begin{bmatrix} 51 & -16 \\ 17 & 21 \end{bmatrix}
\][/tex]
