Given the matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex], compute [tex]\( 2A - 3B \)[/tex].

[tex]\[
A = \left[\begin{array}{rrr}
-1 & 0 & 2 \\
4 & 1 & -1 \\
2 & 0 & 1
\end{array}\right]
\quad
B = \left[\begin{array}{rrr}
2 & 1 & 0 \\
-1 & 0 & 2 \\
4 & -3 & 1
\end{array}\right]
\][/tex]

Enter the element in row 3, column 1 of [tex]\( 2A - 3B \)[/tex].



Answer :

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].