To find the expression [tex]\(-3A + 4B\)[/tex] given matrices [tex]\(A\)[/tex] and [tex]\(B\)[/tex], we need to follow several steps. Let's work through them methodically.
First, let's determine [tex]\(-3A\)[/tex]:
Given matrix [tex]\(A\)[/tex]:
[tex]\[
A = \begin{pmatrix}
2 & -5 \\
-8 & 1
\end{pmatrix}
\][/tex]
We multiply [tex]\(A\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
-3A = -3 \cdot \begin{pmatrix}
2 & -5 \\
-8 & 1
\end{pmatrix}
= \begin{pmatrix}
-3 \cdot 2 & -3 \cdot (-5) \\
-3 \cdot (-8) & -3 \cdot 1
\end{pmatrix}
= \begin{pmatrix}
-6 & 15 \\
24 & -3
\end{pmatrix}
\][/tex]
Next, let's compute [tex]\(4B\)[/tex]:
Given matrix [tex]\(B\)[/tex]:
[tex]\[
B = \begin{pmatrix}
3 & 6 \\
6 & -5
\end{pmatrix}
\][/tex]
We multiply [tex]\(B\)[/tex] by [tex]\(4\)[/tex]:
[tex]\[
4B = 4 \cdot \begin{pmatrix}
3 & 6 \\
6 & -5
\end{pmatrix}
= \begin{pmatrix}
4 \cdot 3 & 4 \cdot 6 \\
4 \cdot 6 & 4 \cdot (-5)
\end{pmatrix}
= \begin{pmatrix}
12 & 24 \\
24 & -20
\end{pmatrix}
\][/tex]
Finally, we add the results of [tex]\(-3A\)[/tex] and [tex]\(4B\)[/tex]:
[tex]\[
-3A + 4B = \begin{pmatrix}
-6 & 15 \\
24 & -3
\end{pmatrix} + \begin{pmatrix}
12 & 24 \\
24 & -20
\end{pmatrix} = \begin{pmatrix}
-6 + 12 & 15 + 24 \\
24 + 24 & -3 + (-20)
\end{pmatrix}
\][/tex]
Calculating each element in the resulting matrix, we get:
[tex]\[
\begin{pmatrix}
-6 + 12 & 15 + 24 \\
24 + 24 & -3 - 20
\end{pmatrix}
= \begin{pmatrix}
6 & 39 \\
48 & -23
\end{pmatrix}
\][/tex]
Therefore, the final result is:
[tex]\[
-3A + 4B = \begin{pmatrix}
6 & 39 \\
48 & -23
\end{pmatrix}
\][/tex]
