To solve the problem, we need to compute [tex]\(3[F] - 2[A]\)[/tex] for the given matrices [tex]\(F\)[/tex] and [tex]\(A\)[/tex].
First, let's write down the matrices:
[tex]\[ [F] = \begin{bmatrix}
1 & 4 & -4 & 7 & -9 \\
3 & -7 & 9 & -5 & -1 \\
-10 & 6 & 5 & -6 & -2
\end{bmatrix} \][/tex]
[tex]\[ [A] = \begin{bmatrix}
8 & 7 & -1 & 10 & 6 \\
4 & -2 & 0 & 1 & -3 \\
-10 & 2 & 9 & -8 & -4
\end{bmatrix} \][/tex]
### Step 1: Calculate [tex]\(3[F]\)[/tex]
We multiply each element of the matrix [tex]\(F\)[/tex] by 3:
[tex]\[ 3[F] = \begin{bmatrix}
3 \cdot 1 & 3 \cdot 4 & 3 \cdot -4 & 3 \cdot 7 & 3 \cdot -9 \\
3 \cdot 3 & 3 \cdot -7 & 3 \cdot 9 & 3 \cdot -5 & 3 \cdot -1 \\
3 \cdot -10 & 3 \cdot 6 & 3 \cdot 5 & 3 \cdot -6 & 3 \cdot -2
\end{bmatrix} \][/tex]
That results in:
[tex]\[ 3[F] = \begin{bmatrix}
3 & 12 & -12 & 21 & -27 \\
9 & -21 & 27 & -15 & -3 \\
-30 & 18 & 15 & -18 & -6
\end{bmatrix} \][/tex]
### Step 2: Calculate [tex]\(2[A]\)[/tex]
We multiply each element of the matrix [tex]\(A\)[/tex] by 2:
[tex]\[ 2[A] = \begin{bmatrix}
2 \cdot 8 & 2 \cdot 7 & 2 \cdot -1 & 2 \cdot 10 & 2 \cdot 6 \\
2 \cdot 4 & 2 \cdot -2 & 2 \cdot 0 & 2 \cdot 1 & 2 \cdot -3 \\
2 \cdot -10 & 2 \cdot 2 & 2 \cdot 9 & 2 \cdot -8 & 2 \cdot -4
\end{bmatrix} \][/tex]
That results in:
[tex]\[ 2[A] = \begin{bmatrix}
16 & 14 & -2 & 20 & 12 \\
8 & -4 & 0 & 2 & -6 \\
-20 & 4 & 18 & -16 & -8
\end{bmatrix} \][/tex]
### Step 3: Calculate [tex]\(3[F] - 2[A]\)[/tex]
Now, we subtract [tex]\(2[A]\)[/tex] from [tex]\(3[F]\)[/tex] element by element:
[tex]\[ 3[F] - 2[A] = \begin{bmatrix}
3 - 16 & 12 - 14 & -12 + 2 & 21 - 20 & -27 - 12 \\
9 - 8 & -21 + 4 & 27 - 0 & -15 - 2 & -3 + 6 \\
-30 + 20 & 18 - 4 & 15 - 18 & -18 + 16 & -6 + 8
\end{bmatrix} \][/tex]
Which simplifies to:
[tex]\[ 3[F] - 2[A] = \begin{bmatrix}
-13 & -2 & -10 & 1 & -39 \\
1 & -17 & 27 & -17 & 3 \\
-10 & 14 & -3 & -2 & 2
\end{bmatrix} \][/tex]
Therefore, the final result of [tex]\(3[F] - 2[A]\)[/tex] is:
[tex]\[ \begin{bmatrix}
-13 & -2 & -10 & 1 & -39 \\
1 & -17 & 27 & -17 & 3 \\
-10 & 14 & -3 & -2 & 2
\end{bmatrix} \][/tex]
