To determine the determinant of the matrix [tex]\( A_x \)[/tex], we need to solve for [tex]\( x \)[/tex] and [tex]\( y \)[/tex] using the given system of linear equations:
[tex]\[
\left[\begin{array}{cc}
5 & -4 \\
3 & 6
\end{array}\right]
\left[\begin{array}{c}
x \\
y
\end{array}\right]
=
\left[\begin{array}{c}
12 \\
66
\end{array}\right]
\][/tex]
Given that [tex]\(\left|A_x\right| = \left|\begin{array}{cc}12 & -4 \\ 66 & 6\end{array}\right|\)[/tex], we calculate the determinant of the matrix [tex]\(\left[\begin{array}{cc}12 & -4 \\ 66 & 6\end{array}\right]\)[/tex].
The determinant of a 2x2 matrix [tex]\(\left[\begin{array}{cc} a & b \\ c & d \end{array}\right]\)[/tex] is given by:
[tex]\[
\text{det} = (a \cdot d) - (b \cdot c)
\][/tex]
Using the entries of the given matrix [tex]\(\left[\begin{array}{cc}12 & -4 \\ 66 & 6\end{array}\right]\)[/tex]:
[tex]\[
a = 12, \quad b = -4, \quad c = 66, \quad d = 6
\][/tex]
Plugging these values into the determinant formula:
[tex]\[
\text{det} = (12 \cdot 6) - (-4 \cdot 66)
\][/tex]
[tex]\[
\text{det} = 72 + 264
\][/tex]
[tex]\[
\text{det} = 336
\][/tex]
Therefore, the determinant [tex]\( \left|A_x\right| \)[/tex] is:
[tex]\[
\left|\begin{array}{cc}12 & -4 \\ 66 & 6\end{array}\right| = 336
\][/tex]
