To solve the problem, we'll follow the specified steps carefully and calculate the required quantities step-by-step.
1. Matrix A and Matrix B:
[tex]\[
A = \begin{pmatrix}
-1 & 0 & 2 \\
4 & 1 & -1 \\
2 & 0 & 1
\end{pmatrix}
\][/tex]
[tex]\[
B = \begin{pmatrix}
2 & 1 & 0 \\
-1 & 0 & 2 \\
4 & -3 & 1
\end{pmatrix}
\][/tex]
2. Compute [tex]\(2A\)[/tex]:
We multiply each element of [tex]\(A\)[/tex] by 2.
[tex]\[
2A = 2 \cdot \begin{pmatrix}
-1 & 0 & 2 \\
4 & 1 & -1 \\
2 & 0 & 1
\end{pmatrix} = \begin{pmatrix}
-2 & 0 & 4 \\
8 & 2 & -2 \\
4 & 0 & 2
\end{pmatrix}
\][/tex]
3. Compute [tex]\(3B\)[/tex]:
We multiply each element of [tex]\(B\)[/tex] by 3.
[tex]\[
3B = 3 \cdot \begin{pmatrix}
2 & 1 & 0 \\
-1 & 0 & 2 \\
4 & -3 & 1
\end{pmatrix} = \begin{pmatrix}
6 & 3 & 0 \\
-3 & 0 & 6 \\
12 & -9 & 3
\end{pmatrix}
\][/tex]
4. Compute [tex]\(2A - 3B\)[/tex]:
We subtract the matrix [tex]\(3B\)[/tex] from [tex]\(2A\)[/tex], element-wise.
[tex]\[
2A - 3B = \begin{pmatrix}
-2 & 0 & 4 \\
8 & 2 & -2 \\
4 & 0 & 2
\end{pmatrix} - \begin{pmatrix}
6 & 3 & 0 \\
-3 & 0 & 6 \\
12 & -9 & 3
\end{pmatrix} = \begin{pmatrix}
-2 - 6 & 0 - 3 & 4 - 0 \\
8 + 3 & 2 - 0 & -2 - 6 \\
4 - 12 & 0 + 9 & 2 - 3
\end{pmatrix}
\][/tex]
Simplifying further:
[tex]\[
2A - 3B = \begin{pmatrix}
-8 & -3 & 4 \\
11 & 2 & -8 \\
-8 & 9 & -1
\end{pmatrix}
\][/tex]
5. Element in row 3, column 1:
In matrix notation, row 3, column 1 corresponds to the element at (3,1).
From our calculated matrix [tex]\(2A - 3B\)[/tex]:
[tex]\[
\begin{pmatrix}
-8 & -3 & 4 \\
11 & 2 & -8 \\
-8 & 9 & -1
\end{pmatrix}_{(3,1)} = -8
\][/tex]
Thus, the element in row 3, column 1 of [tex]\(2A - 3B\)[/tex] is [tex]\(-8\)[/tex].
