To determine the result of multiplying the matrix [tex]\( C \)[/tex] by 4, we start with matrix [tex]\( C \)[/tex]:
[tex]\[
C = \left[\begin{array}{ccc}
12 & 0 & \frac{3}{2} \\
1 & -6 & 7
\end{array}\right]
\][/tex]
We multiply each element of the matrix [tex]\( C \)[/tex] by 4:
[tex]\[
4 \times C = 4 \times \left[\begin{array}{ccc}
12 & 0 & \frac{3}{2} \\
1 & -6 & 7
\end{array}\right]
\][/tex]
We compute each element of the resulting matrix [tex]\( 4C \)[/tex] as follows:
- The element in the first row, first column: [tex]\( 4 \times 12 = 48 \)[/tex]
- The element in the first row, second column: [tex]\( 4 \times 0 = 0 \)[/tex]
- The element in the first row, third column: [tex]\( 4 \times \frac{3}{2} = 4 \times 1.5 = 6 \)[/tex]
- The element in the second row, first column: [tex]\( 4 \times 1 = 4 \)[/tex]
- The element in the second row, second column: [tex]\( 4 \times (-6) = -24 \)[/tex]
- The element in the second row, third column: [tex]\( 4 \times 7 = 28 \)[/tex]
Thus, the matrix [tex]\( 4C \)[/tex] is:
[tex]\[
4C = \left[\begin{array}{ccc}
48 & 0 & 6 \\
4 & -24 & 28
\end{array}\right]
\][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{\left[\begin{array}{rrr}48 & 0 & 6 \\ 4 & -24 & 28\end{array}\right]} \][/tex]
This corresponds to option A.
