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Let [tex]A=\left[\begin{array}{rr}2 & -5 \\ -8 & 1\end{array}\right][/tex] and [tex]B=\left[\begin{array}{rr}3 & 6 \\ 6 & -5\end{array}\right][/tex].

Find [tex]-3A + 4B[/tex].

[tex]-3A + 4B =[/tex]


Sagot :

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]
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