To determine the matrix product [tex]\( FC \)[/tex], we need to perform the matrix multiplication of [tex]\( F \)[/tex] and [tex]\( C \)[/tex].
The matrix [tex]\( F \)[/tex] is given as:
[tex]\[ F = \begin{bmatrix}
-2 & 0 \\
0 & 8 \\
2 & 1
\end{bmatrix} \][/tex]
The matrix [tex]\( C \)[/tex] is given as:
[tex]\[ C = \begin{bmatrix}
12 & 0 & \frac{3}{2} \\
1 & -6 & 7
\end{bmatrix} \][/tex]
We will multiply these matrices step-by-step to find the resulting matrix [tex]\( FC \)[/tex].
The multiplication of two matrices involves taking the dot product of rows of the first matrix with the columns of the second matrix. For example, if [tex]\( A \)[/tex] is a [tex]\( m \times n \)[/tex] matrix and [tex]\( B \)[/tex] is a [tex]\( n \times p \)[/tex] matrix, their product [tex]\( AB \)[/tex] is a [tex]\( m \times p \)[/tex] matrix. Each element [tex]\( (i, j) \)[/tex] of the resulting matrix is computed as:
[tex]\[ (AB)_{ij} = \sum_{k=1}^{n} A_{ik} \times B_{kj} \][/tex]
For this problem, the element calculations are:
1. For the first row of [tex]\( F \)[/tex] and first column of [tex]\( C \)[/tex]:
[tex]\[ (-2 \times 12) + (0 \times 1) = -24 \][/tex]
2. For the first row of [tex]\( F \)[/tex] and second column of [tex]\( C \)[/tex]:
[tex]\[ (-2 \times 0) + (0 \times -6) = 0 \][/tex]
3. For the first row of [tex]\( F \)[/tex] and third column of [tex]\( C \)[/tex]:
[tex]\[ (-2 \times 1.5) + (0 \times 7) = -3 \][/tex]
4. For the second row of [tex]\( F \)[/tex] and first column of [tex]\( C \)[/tex]:
[tex]\[ (0 \times 12) + (8 \times 1) = 8 \][/tex]
5. For the second row of [tex]\( F \)[/tex] and second column of [tex]\( C \)[/tex]:
[tex]\[ (0 \times 0) + (8 \times -6) = -48 \][/tex]
6. For the second row of [tex]\( F \)[/tex] and third column of [tex]\( C \)[/tex]:
[tex]\[ (0 \times 1.5) + (8 \times 7) = 56 \][/tex]
7. For the third row of [tex]\( F \)[/tex] and first column of [tex]\( C \)[/tex]:
[tex]\[ (2 \times 12) + (1 \times 1) = 25 \][/tex]
8. For the third row of [tex]\( F \)[/tex] and second column of [tex]\( C \)[/tex]:
[tex]\[ (2 \times 0) + (1 \times -6) = -6 \][/tex]
9. For the third row of [tex]\( F \)[/tex] and third column of [tex]\( C \)[/tex]:
[tex]\[ (2 \times 1.5) + (1 \times 7) = 10 \][/tex]
So when we combine all these values, we get:
[tex]\[ FC = \begin{bmatrix}
-24 & 0 & -3 \\
8 & -48 & 56 \\
25 & -6 & 10
\end{bmatrix} \][/tex]
Thus, the correct answer is:
D. [tex]\(\begin{bmatrix}
-24 & 0 & -3 \\
8 & -48 & 56 \\
25 & -6 & 10
\end{bmatrix}\)[/tex]
